tag:blogger.com,1999:blog-20446223125735189702012-02-16T18:42:10.108+07:00The love makes us stronger!Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.comBlogger1891100tag:blogger.com,1999:blog-2044622312573518970.post-42730598143864170972010-04-01T00:00:00.007+07:002010-04-01T07:10:35.327+07:00<div style="text-align: justify;">I am writing a new inequality book named "Rèn luyện kĩ năng sử dụng bất đẳng thức Cauchy - Schwarz thông qua các bài toán." In English, it means "Improving your skills of using the Cauchy - Schwarz Inequality through problems." In the book, I presented all techniques I know about the Cauchy - Schwarz Inequality. I hope that the readers can improve their skills of using the Cauchy - Schwarz Inequality after reading this book. The book is written in Vietnamese and it will appear soon. Here is the cover (it was designed by me):<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/_ZZl4BL5SH3E/S7PkUIAN0VI/AAAAAAAAAFM/1jMpvHd7z_Q/s1600/BIA.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/_ZZl4BL5SH3E/S7PkUIAN0VI/AAAAAAAAAFM/1jMpvHd7z_Q/s320/BIA.jpg" /></a></div></div><br /><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/grIvQXrAUJ" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/grIvQXrAUJ" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4273059814386417097?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com10tag:blogger.com,1999:blog-2044622312573518970.post-62111417332590549732010-03-20T08:39:00.001+07:002010-03-20T08:49:58.402+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%5E%7B2%7D-bd%7D%7Bb+2c+d%7D+%5Cfrac%7Bb%5E%7B2%7D-ca%7D%7Bc+2d+a%7D+%5Cfrac%7Bc%5E%7B2%7D-db%7D%7Bd+2a+b%7D+%5Cfrac%7Bd%5E%7B2%7D-ac%7D%7Ba+2b+c%7D%5Cgeq%200.$$" title="$$\frac{a^{2}-bd}{b+2c+d}+\frac{b^{2}-ca}{c+2d+a}+\frac{c^{2}-db}{d+2a+b}+\frac{d^{2}-ac}{a+2b+c}\geq 0.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMTc0YjgxYTYtYTZkMS00ZmYyLWEwM2EtOGUyMTZkOTgyZmIx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/3V3oKaTmvR" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/3V3oKaTmvR" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6211141733259054973?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com1tag:blogger.com,1999:blog-2044622312573518970.post-50353535245086528342010-03-19T20:18:00.001+07:002010-03-19T21:02:26.402+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Csum%20%5Cfrac%7Bb%5E3+c%5E3%7D%7Ba%5E2+bc%7D%5Cge%202%5Csum%20%5Cfrac%7Ba%5E2%7D%7Bb+c%7D." title="\sum \frac{b^3+c^3}{a^2+bc}\ge 2\sum \frac{a^2}{b+c}." /></i></div><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMjEwOWQ1ZjQtZmJmNi00YThkLThiNjAtYWUwYTJmOGJjYTVh&amp;hl=en">Download here.</a><br /><br /><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/c87_qUY2Vl" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/c87_qUY2Vl" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5035353524508652834?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-69915286527677894602010-03-18T16:12:00.001+07:002010-03-18T16:15:16.237+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be positive real numbers such that <img src="http://latex.codecogs.com/gif.latex?a+b+c+d=4." title="a+b+c+d=4." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7Bab%7D%7B4-cd%7D+4%5Cleft%28%20%5Cfrac%7Bac%7D%7B4-bd%7D+%5Cfrac%7Bbd%7D%7B4-ac%7D%5Cright%29%20%5Cleq%204.$$" title="$$\sum \frac{ab}{4-cd}+4\left( \frac{ac}{4-bd}+\frac{bd}{4-ac}\right) \leq 4.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNjYzNWQyZDYtZmE5NC00ZDE3LWEyNzctYjAxZTY2MDA5ZTcw&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/zpJJDttHkv" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/zpJJDttHkv" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6991528652767789460?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-39579007192355050432010-03-18T14:18:00.001+07:002010-03-18T14:27:23.876+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7B%28b+c%29%28b%5E%7B2%7D+c%5E%7B2%7D%29%7D%7Ba%5E%7B2%7D+bc%7D%5Cgeq%20%5Cfrac%7B6%28a%5E%7B3%7D+b%5E%7B3%7D+c%5E%7B3%7D%29%7D%7Ba%5E%7B2%7D+b%5E%7B2%7D+c%5E%7B2%7D%7D.$$" title="$$\sum \frac{(b+c)(b^{2}+c^{2})}{a^{2}+bc}\geq \frac{6(a^{3}+b^{3}+c^{3})}{a^{2}+b^{2}+c^{2}}.$$" /></i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXY2UzMjNlMTYtMjhjMC00ODcyLWFkMTYtYWEzM2EyYTFkMGY5&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/qnvMKyC2ny" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/qnvMKyC2ny" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3957900719235505043?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com1tag:blogger.com,1999:blog-2044622312573518970.post-22492261128750847452010-03-15T09:33:00.002+07:002010-03-15T09:35:19.475+07:00<i>Let <img src="http://latex.codecogs.com/gif.latex?n%5Cgeq%202" title="n\geq 2" /> be a positive integer. Positive real numbers <img src="http://latex.codecogs.com/gif.latex?a_%7B1%7D,a_%7B2%7D,%5Cldots%20,a_%7Bn%7D" title="a_{1},a_{2},\ldots ,a_{n}" /> and real numbers <img src="http://latex.codecogs.com/gif.latex?x_%7B1%7D,x_%7B2%7D,%5Cldots%20,x_%7Bn%7D" title="x_{1},x_{2},\ldots ,x_{n}" /> satisfy</i><br /><div style="text-align: center;"><i> <img src="http://latex.codecogs.com/gif.latex?a_%7B1%7Dx_%7B1%7D+a_%7B2%7Dx_%7B2%7D+%5Ccdots%20+a_%7Bn%7Dx_%7Bn%7D=0." title="a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=0." />&nbsp;</i></div><i>Prove that</i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum_%7B1%5Cleq%20i%3Cj%5Cleq%20n%7Dx_%7Bi%7Dx_%7Bj%7D%7Ca_%7Bi%7D-a_%7Bi%7D%7C%5Cleq%200.$$" title="$$\sum_{1\leq i&lt;j\leq n}x_{i}x_{j}|a_{i}-a_{i}|\leq 0.$$" /></i></div><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTYzM2ExZGItMmUwNy00ZDhiLWI0MzEtNDdkYjQ5MjJjZWEw&amp;hl=en">Download here.</a><br /><br /><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/QQ4iCqRlfU" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/QQ4iCqRlfU" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2249226112875084745?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-34913957825624945152010-03-15T08:00:00.001+07:002010-03-15T08:02:04.123+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be positive real numbers such that <img src="http://latex.codecogs.com/gif.latex?a%5Cgeq%20b%5Cgeq%20c%5Cgeq%20d" title="a\geq b\geq c\geq d" /> and<br /></i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$3%28a%5E%7B2%7D+b%5E%7B2%7D+c%5E%7B2%7D+d%5E%7B2%7D%29=%28a+b+c+d%29%5E%7B2%7D.$$" title="$$3(a^{2}+b^{2}+c^{2}+d^{2})=(a+b+c+d)^{2}.$$" /></i></div><i> Prove that</i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Cfrac%7Ba+c%7D%7Bb+d%7D%20%5Cle%20%5Cfrac%7B7+2%5Csqrt%7B6%7D%7D%7B5%7D." title="\frac{a+c}{b+d} \le \frac{7+2\sqrt{6}}{5}." /> </i></div></div><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMDE4OWZlYWEtMTJhNC00YWNiLWJlMTAtZmEzNjI1ODZlMDA1&amp;hl=en">Download here.</a><br /><br /><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/phk4Sk2PmM" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/phk4Sk2PmM" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3491395782562494515?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-30563455543861881432010-03-14T21:29:00.001+07:002010-03-14T21:31:45.544+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7B1%7D%7B2a%5E%7B2%7D+bc%7D+%5Cfrac%7B1%7D%7B2b%5E%7B2%7D+ca%7D+%5Cfrac%7B1%7D%7B2c%5E%7B2%7D+ab%7D%5Cleq%20%5Cleft%28%20%5Cfrac%7Ba+b+c%7D%7Bab+bc+ca%7D%5Cright%29%20%5E%7B2%7D.$$" title="$$\frac{1}{2a^{2}+bc}+\frac{1}{2b^{2}+ca}+\frac{1}{2c^{2}+ab}\leq \left( \frac{a+b+c}{ab+bc+ca}\right) ^{2}.$$" /></i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNzM5MDIyMWEtYmMzNi00MDFmLTg1MmYtZTU0ZTAzMTMxZjVj&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/NbjZlj820E" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/NbjZlj820E" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3056345554386188143?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-88139955168256013502010-03-14T19:15:00.004+07:002010-03-14T19:18:18.017+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be nonnegative real numbers such that <img src="http://latex.codecogs.com/gif.latex?a%5Cgeq%20b%5Cgeq%20c%5Cgeq%20d" title="a\geq b\geq c\geq d" /> and<br /></i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$2%28a%5E%7B2%7D+b%5E%7B2%7D+c%5E%7B2%7D+d%5E%7B2%7D%29=%28a+b+c+d%29%5E%7B2%7D.$$" title="$$2(a^{2}+b^{2}+c^{2}+d^{2})=(a+b+c+d)^{2}.$$" /></i></div><i> Prove that</i><br /><div style="text-align: center;"><img src="http://latex.codecogs.com/gif.latex?a%20%5Cge%20b%20+3c%20+%5Cleft%282%5Csqrt%7B3%7D-1%20%5Cright%20%29d." title="a \ge b +3c +\left(2\sqrt{3}-1 \right )d." /></div></div><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZjAxMjAwNjctZDQzOS00NTkxLWFjNDctODQ5ZmMzODk3MDhk&amp;hl=en">Download here.</a><br /><br /><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/MtjmSe05t0" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/MtjmSe05t0" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8813995516825601350?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-78532722165165293492010-03-14T17:56:00.003+07:002010-03-14T19:13:24.694+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be nonnegative real numbers such that <img src="http://latex.codecogs.com/gif.latex?a%5Cgeq%20b%5Cgeq%20c%5Cgeq%20d" title="a\geq b\geq c\geq d" /> and<br /></i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$3%28a%5E%7B2%7D+b%5E%7B2%7D+c%5E%7B2%7D+d%5E%7B2%7D%29=%28a+b+c+d%29%5E%7B2%7D.$$" title="$$3(a^{2}+b^{2}+c^{2}+d^{2})=(a+b+c+d)^{2}.$$" /></i></div><i> Prove that</i></div><ul><li><i><img src="http://latex.codecogs.com/gif.latex?%5Cbegin%7Baligned%7D%20&amp;%28a%29%5C%20%5C%20%5C%20a%5Cleq%203b;%5C%5C%20&amp;%28b%29%5C%20%5C%20%5C%20a%5Cleq%204c;%5C%5C%20&amp;%28c%29%5C%20%5C%20%5C%20b%5Cleq%20%5Cleft%28%202+%5Csqrt%7B3%7D%5Cright%29%20c.%5Cend%7Baligned%7D" title="\begin{aligned} &amp;(a)\ \ \ a\leq 3b;\\ &amp;(b)\ \ \ a\leq 4c;\\ &amp;(c)\ \ \ b\leq \left( 2+\sqrt{3}\right) c.\end{aligned}" /><br /></i></li></ul><div style="text-align: justify;"></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof.</b></i> <a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMThkNzczYzktOTliNi00ODAxLWFjMDMtOGIxMzc4OTZjMDRk&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/E6HKFnHDCA" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/E6HKFnHDCA" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7853272216516529349?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-30110782920300532132010-03-14T01:31:00.003+07:002010-03-14T01:45:51.123+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Bbc%7D%7Ba%5E%7B2%7D+bc%7D+%5Cfrac%7Bca%7D%7Bb%5E%7B2%7D+ca%7D+%5Cfrac%7Bab%7D%7Bc%5E%7B2%7D+ab%7D+%5Cfrac%7B2%28ab+bc+ca%29%7D%7B%28a+b+c%29%5E%7B2%7D%7D%5Cleq%20%5Cfrac%7B13%7D%7B6%7D.$$" title="$$\frac{bc}{a^{2}+bc}+\frac{ca}{b^{2}+ca}+\frac{ab}{c^{2}+ab}+\frac{2(ab+bc+ca)}{(a+b+c)^{2}}\leq \frac{13}{6}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXY2UyNWJiOTUtZTQyOS00NzNkLWFkYzUtMjk0ZTkxY2NlYjdi&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/DPtF3iP0Wn" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/DPtF3iP0Wn" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3011078292030053213?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-7193437670209392442010-03-12T23:11:00.002+07:002010-03-12T23:16:53.366+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers such that <img src="http://latex.codecogs.com/gif.latex?ab+bc+ca=3." title="ab+bc+ca=3." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7B1%7D%7B%28b+c%29%5E%7B2%7D%7D%5Cgeq%20%5Cfrac%7B3%7D%7B4%7D+%5Cfrac%7B3%7D%7B2%7D%5Ccdot%20%5Cfrac%7B%28a-1%29%28b-1%29%28c-1%29%7D%7B%28b+c%29%28c+a%29%28a+b%29%7D.$$" title="$$\sum \frac{1}{(b+c)^{2}}\geq \frac{3}{4}+\frac{3}{2}\cdot \frac{(a-1)(b-1)(c-1)}{(b+c)(c+a)(a+b)}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGJmZTI5MzctMGVlOC00YzkyLThiMzMtZTA2MzgwMzQyZmU4&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><a name='more'></a></div><div style="text-align: justify;"><br /></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/CoNcUinDlx" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/CoNcUinDlx" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-719343767020939244?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-43711139495372486452010-03-12T19:24:00.002+07:002010-03-12T19:32:58.228+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers such that <img src="http://latex.codecogs.com/gif.latex?a+b+c=8." title="a+b+c=8." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Csqrt%7Ba%28b+c%29%7D%5Cgeq%20%5Csqrt[4]%7B%28a%5E%7B2%7D+bc%29%28b%5E%7B2%7D+ca%29%28c%5E%7B2%7D+ab%29%7D.$$" title="$$\sum \sqrt{a(b+c)}\geq \sqrt[4]{(a^{2}+bc)(b^{2}+ca)(c^{2}+ab)}.$$" /></i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZGU3NGEyYzEtNzE0OC00YjAyLTk3ZTUtMDFkMzY4ZDhjMmM5&amp;hl=en">Download here.</a><i></i></div><div style="text-align: justify;"><i><br /></i></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/qvmofm_OfJ" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/qvmofm_OfJ" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4371113949537248645?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-83862840835053399392010-03-12T18:42:00.001+07:002010-03-12T18:50:48.076+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Bb%7D%7Ba%7D+%5Cfrac%7Bc%7D%7Bb%7D+%5Cfrac%7Ba%7D%7Bc%7D%20%5Cge%20%5Cfrac%7Ba%5E2+bc%7D%7Bb%5E2+ca%7D+%5Cfrac%7Bb%5E2+ca%7D%7Bc%5E2+ab%7D+%5Cfrac%7Bc%5E2+ab%7D%7Ba%5E2+bc%7D.$$" title="$$\frac{b}{a}+\frac{c}{b}+\frac{a}{c} \ge \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}.$$" /></i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMzAzODhhYzgtNmQ1OS00NjY5LTkxYjctMTJmMDY4ZjcxNDAz&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/nWmibS8SaP" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/nWmibS8SaP" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8386284083505339939?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-86420418793918220912010-03-11T09:15:00.004+07:002010-03-11T09:17:57.365+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Bb+c%7D%7Ba%5E%7B2%7D+bc%7D+%5Cfrac%7Bc+a%7D%7Bb%5E%7B2%7D+ca%7D+%5Cfrac%7Ba+b%7D%7Bc%5E%7B2%7D+ab%7D%5Cleq%20%5Cfrac%7Ba+b+c%7D%7B%5Csqrt[3]%7Ba%5E%7B2%7Db%5E%7B2%7Dc%5E%7B2%7D%7D%7D.$$" title="$$\frac{b+c}{a^{2}+bc}+\frac{c+a}{b^{2}+ca}+\frac{a+b}{c^{2}+ab}\leq \frac{a+b+c}{\sqrt[3]{a^{2}b^{2}c^{2}}}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOWYxZDcyY2YtMmQxYS00OWNmLWE1MjktNzNhY2FlNjg0MzEw&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Go6_VBV7gj" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Go6_VBV7gj" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8642041879391822091?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com1tag:blogger.com,1999:blog-2044622312573518970.post-71342141367785606802010-03-11T08:50:00.003+07:002010-03-11T08:56:40.435+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. If <img src="http://latex.codecogs.com/gif.latex?n%3E0," title="n&gt;0," /> then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csqrt[n]%7B%5Cfrac%7B%28a+b%29%28b+c%29%28c+a%29%7D%7B8abc%7D%7D%5Cgeq%20%5Cfrac%7B2%7D%7Bn%7D%5Csum%5Cfrac%7Ba%5E%7B2%7D%7D%7Ba%5E%7B2%7D+bc%7D%20+%5Cfrac%7Bn-3%7D%7Bn%7D.$$" title="$$\sqrt[n]{\frac{(a+b)(b+c)(c+a)}{8abc}}\geq \frac{2}{n}\sum\frac{a^{2}}{a^{2}+bc} +\frac{n-3}{n}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNTJkZDMyMjEtMzkzYi00YzdlLWFjOGUtYWIxZmU3ZWMwYzRh&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/IFUHpWtbK3" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/IFUHpWtbK3" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7134214136778560680?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-64380110051333300572010-03-11T07:50:00.004+07:002010-03-11T07:52:51.410+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7Bb+c%7D%7Ba%5E%7B2%7D+bc%7D%5Ccdot%20%5Cfrac%7Bc+a%7D%7Bb%5E%7B2%7D+ca%7D%5Cgeq%20%5Csum%20%5Cfrac%7B2%7D%7Ba%5E%7B2%7D+b%5E%7B2%7D%7D.$$" title="$$\sum \frac{b+c}{a^{2}+bc}\cdot \frac{c+a}{b^{2}+ca}\geq \sum \frac{2}{a^{2}+b^{2}}.$$" /></i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZDcwNmExZDAtYWM1ZC00Mzc2LWJjZWUtOTlhOTY3Nzg2MjU1&amp;hl=en">Download here.</a><i></i><br /><i></i><br /><a name='more'></a><i><br /></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/ihzrkjbV74" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/ihzrkjbV74" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6438011005133330057?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-40274141122627888702010-03-10T14:44:00.002+07:002010-03-10T14:47:14.842+07:00<div style="text-align: justify;">The site <a href="http://maths.vn/">http://maths.vn</a> has just organized successfully an Inequality Contest for Juniors. Visitors can download the Problems and Solutions of Second Round here:</div><ul><li><a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/free%20public/De%20Thi%20THCS.pdf">Problems.</a></li><li><a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/free%20public/dapan.pdf">Solutions.</a></li></ul><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/vtNXiKxnDD" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/vtNXiKxnDD" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4027414112262788870?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-48534149854910325072010-03-10T14:32:00.002+07:002010-03-10T14:35:16.924+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. If <img src="http://latex.codecogs.com/gif.latex?k%5Cgeq%202," title="k\geq 2," /> then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7Ba%7D%7Ba%5E%7B2%7D+kbc%7D%5Cleq%20%5Cfrac%7B%5Cleft%28%5Csum%20a%20%5Cright%20%29%5Cleft[%28k-2%29%5Csum%20a%5E2+%285-k%29%5Csum%20ab%5Cright]%7D%7B%28k+1%29%5Cleft%28%5Csum%20ab%20%5Cright%20%29%5E2%7D.$$" title="$$\sum \frac{a}{a^{2}+kbc}\leq \frac{\left(\sum a \right )\left[(k-2)\sum a^2+(5-k)\sum ab\right]}{(k+1)\left(\sum ab \right )^2}.$$" /></i></div><div style="text-align: justify;"><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNTBhMWNkYTQtZDU1YS00NzNiLWI5OGQtMzQ2NTYzYWU2NDcx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/yrr_V4U1LK" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/yrr_V4U1LK" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4853414985491032507?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-80878547034892238492010-03-10T12:45:00.002+07:002010-03-10T12:51:50.315+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%28b+c%29%7D%7Ba%5E2+bc%7D+%5Cfrac%7Bb%28c+a%29%7D%7Bb%5E2+ca%7D+%5Cfrac%7Bc%28a+b%29%7D%7Bc%5E2+ab%7D%20%5Cle%20%5Cfrac%7Ba+b+c%7D%7B%5Csqrt[3]%7Babc%7D%7D.$$" title="$$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab} \le \frac{a+b+c}{\sqrt[3]{abc}}.$$" /></i> </div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGI1MWFiYzQtNTI5OC00MzM5LWE3ZDUtYTFmYTI2ZWFjMzgy&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/dIgJloCM6i" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/dIgJloCM6i" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8087854703489223849?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-12598696007305329052010-03-09T07:31:00.003+07:002010-03-09T16:02:02.141+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7B%28a%5E%7B2%7D+bc%29%28b%5E%7B2%7D+ca%29%28c%5E%7B2%7D+ab%29%7D%7B%28a+b%29%28b+c%29%28c+a%29%7D%5Cgeq%20%5Cfrac%7Ba%5E%7B2%7Db%5E%7B2%7D+b%5E%7B2%7Dc%5E%7B2%7D+c%5E%7B2%7Da%5E%7B2%7D%7D%7Ba+b+c%7D.$$" title="$$\frac{(a^{2}+bc)(b^{2}+ca)(c^{2}+ab)}{(a+b)(b+c)(c+a)}\geq \frac{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}{a+b+c}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZDQwZDVkMDEtODllZS00NzdkLTkzYzUtY2RkNGY3ZGMyYzI3&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Ag1er42L6m" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Ag1er42L6m" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1259869600730532905?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-82553876182701625702010-03-07T20:31:00.002+07:002010-03-07T20:47:08.611+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%28b+c%29%7D%7Ba%5E%7B2%7D+bc%7D+%5Cfrac%7Bb%28c+a%29%7D%7Bb%5E%7B2%7D+ca%7D+%5Cfrac%7Bc%28a+b%29%7D%7Bc%5E%7B2%7D+ab%7D%5Cleq%20%5Cfrac%7B%28a+b+c%29%5E%7B2%7D%7D%7Bab+bc+ca%7D.$$" title="$$\frac{a(b+c)}{a^{2}+bc}+\frac{b(c+a)}{b^{2}+ca}+\frac{c(a+b)}{c^{2}+ab}\leq \frac{(a+b+c)^{2}}{ab+bc+ca}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYjVmNzY2Y2EtY2MwZi00YjYyLWE1MWMtNDc3NTdmZGU0YmNj&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Uvh9zab0Ni" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Uvh9zab0Ni" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8255387618270162570?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-16627122553854005422010-03-07T18:45:00.005+07:002010-03-07T18:52:26.452+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7B%28a%5E%7B2%7D+bc%29%28b%5E%7B2%7D+ca%29%28c%5E%7B2%7D+ab%29%7D%7B%28a+b%29%28b+c%29%28c+a%29%7D%5Cgeq%20%5Cfrac%7B%28ab+bc+ca%29%5E%7B2%7D%7D%7B3%28a+b+c%29%7D.$$" title="$$\frac{(a^{2}+bc)(b^{2}+ca)(c^{2}+ab)}{(a+b)(b+c)(c+a)}\geq \frac{(ab+bc+ca)^{2}}{3(a+b+c)}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMGNmNjgxZTMtOGY5ZC00MWUyLTkwMGMtZWFkMDgyZTJjYWE0&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/ZpbJpax_pt" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/ZpbJpax_pt" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1662712255385400542?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-82316170814716616712010-03-06T02:13:00.005+07:002010-03-06T07:01:00.664+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?0%20%5Cle%20x,y,z%20%5Cle%20%5Cpi" title="0 \le x,y,z \le \pi" /> such that <img src="http://latex.codecogs.com/gif.latex?x+y+z=%5Cpi." title="x+y+z=\pi." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Ctan%20%5Cfrac%7Bx%7D%7B4%7D%5Ctan%20%5Cfrac%7By%7D%7B4%7D+%5Ctan%20%5Cfrac%7By%7D%7B4%7D%5Ctan%20%5Cfrac%7Bz%7D%7B4%7D+%5Ctan%20%5Cfrac%7Bz%7D%7B4%7D%5Ctan%20%5Cfrac%7Bx%7D%7B4%7D%20%5Cle%203%5Cleft%282-%5Csqrt%7B3%7D%5Cright%29%5E2.$$" title="$$\tan \frac{x}{4}\tan \frac{y}{4}+\tan \frac{y}{4}\tan \frac{z}{4}+\tan \frac{z}{4}\tan \frac{x}{4} \le 3\left(2-\sqrt{3}\right)^2.$$" /></i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNmIwNzk2MGEtZmRiMC00NmQwLWEwN2UtNDQ1OWJlMjMxNTFj&amp;hl=en">Download here.</a><i></i><br /><i></i><br /><a name='more'></a><i><br /></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/2LCIAlfSyx" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/2LCIAlfSyx" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8231617081471661671?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-52123609019814748162010-03-06T01:17:00.002+07:002010-03-06T01:20:18.130+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%5E%7B2%7D%7D%7Ba%5E%7B2%7D+bc%7D+%5Cfrac%7Bb%5E%7B2%7D%7D%7Bb%5E%7B2%7D+ca%7D+%5Cfrac%7Bc%5E%7B2%7D%7D%7Bc%5E%7B2%7D+ab%7D%5Cleq%20%5Cfrac%7Ba+b+c%7D%7B2%5Csqrt[3]%7Babc%7D%7D.$$" title="$$\frac{a^{2}}{a^{2}+bc}+\frac{b^{2}}{b^{2}+ca}+\frac{c^{2}}{c^{2}+ab}\leq \frac{a+b+c}{2\sqrt[3]{abc}}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNjQ5OGMyM2MtMTYzZS00YmRjLThlOTMtMmM1MDU2YzM1N2Jk&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/mYK2vcWYNs" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/mYK2vcWYNs" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5212360901981474816?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-4138351446152705632010-03-04T11:39:00.005+07:002010-03-15T16:22:01.004+07:00<ul><li><b>1991</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201991.rar">Full volume</a>.&nbsp; <b><br /></b></li><li><b>1992</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201992.rar">Full volume</a>.&nbsp; <b><br /></b></li><li><b>1993</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201993.rar">Full volume</a>. </li><li><b>1994</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201994.rar">Full volume</a>.&nbsp; <b><br /></b></li><li><b>1995</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201995.rar">Full volume</a>. </li><li><b>1996</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201996.rar">Full volume</a>. </li><li><b>1997</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201997.rar">Full volume</a>. </li><li><b>1998</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201998.rar">Full volume</a>. </li><li><b>1999</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%201999.rar">Full volume</a>.&nbsp; </li><li><b>2000</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202000.rar">Full volume</a>.&nbsp; </li><li><b>2001</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202001.rar">Full volume</a>.&nbsp; </li><li><b>2002</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202002.rar">Full volume</a>.&nbsp; </li><li><b>2003</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202003.rar">Full volume</a>. </li><li><b>2004</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202004.rar">Full volume</a>. </li><li><b>2005</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202005.rar">Full volume</a>. </li><li><b>2006</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202006.rar">Full volume</a>.</li><li><b>2007</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202007.rar">Full volume</a>.</li><li><b>2008</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202008.rar">Full volume</a>.</li><li><b>2009</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/crux%202009.rar">Full volume</a>.</li><li><b>2010</b> - <a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Crux%20Mathematicorum/cruxv36n1.pdf">No. 1</a>. </li></ul><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Ouq5xMiEcy" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Ouq5xMiEcy" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-413835144615270563?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com1tag:blogger.com,1999:blog-2044622312573518970.post-51255176547241547122010-03-04T09:25:00.005+07:002010-03-04T09:32:41.647+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7Ba%7D%7B2a%5E%7B2%7D+bc+ca%7D%5Cgeq%202abc%5Csum%20%5Cfrac%7B1%7D%7B%28a%5E%7B2%7D+b%5E%7B2%7D%29%283c%5E%7B2%7D+a%5E%7B2%7D%29%7D.$$" title="$$\sum \frac{a}{2a^{2}+bc+ca}\geq 2abc\sum \frac{1}{(a^{2}+b^{2})(3c^{2}+a^{2})}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYWQ1OTM0M2QtNDVkYi00OGI2LTk2ZjYtMTI1NTEzMWQ3ZmQ1&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/BrdHGU5K8b" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/BrdHGU5K8b" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5125517654724154712?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-74130790447740147872010-03-04T08:33:00.004+07:002010-03-04T08:41:42.380+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a_%7B1%7D,a_%7B2%7D,%5Cldots%20,a_%7Bn%7D" title="a_{1},a_{2},\ldots ,a_{n}" /> be nonzero real numbers sucht that </i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cbegin%7Barray%7D%7Bc%7D%20a_%7B1%7D,a_%7B2%7D,%5Cldots%20,a_%7Bn%7D%5Cgeq%20n-1-%5Csqrt%7Bn%5E%7B2%7D-n+1%7D,%5C%5C%5C%5C%20a_%7B1%7D+a_%7B2%7D+%5Ccdots%20+a_%7Bn%7D=n.%5Cend%7Barray%7D$$" title="$$\begin{array}{c} a_{1},a_{2},\ldots ,a_{n}\geq n-1-\sqrt{n^{2}-n+1},\\\\ a_{1}+a_{2}+\cdots +a_{n}=n.\end{array}$$" /></i></div><div style="text-align: justify;"><i>Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7B1%7D%7Ba_%7B1%7D%5E%7B2%7D%7D+%5Cfrac%7B1%7D%7Ba_%7B2%7D%5E%7B2%7D%7D+%5Ccdots%20+%5Cfrac%7B1%7D%7Ba_%7Bn%7D%5E%7B2%7D%7D%5Cgeq%20n.$$" title="$$\frac{1}{a_{1}^{2}}+\frac{1}{a_{2}^{2}}+\cdots +\frac{1}{a_{n}^{2}}\geq n.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGExNzM5OTMtMDZkYS00MDNhLThmNjAtMWE1YzllMDA3MDc5&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/L1_gPT0PwO" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/L1_gPT0PwO" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7413079044774014787?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-71957965868474493002010-03-04T07:14:00.003+07:002010-03-04T07:21:43.924+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers such that $a+b+c=3.$ Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%7D%7B1+a+b%5E%7B2%7D%7D+%5Cfrac%7Bb%7D%7B1+b+c%5E%7B2%7D%7D+%5Cfrac%7Bc%7D%7B1+c+a%5E%7B2%7D%7D%5Cgeq%20%5Cfrac%7B3%7D%7B4%7D.$$" title="$$\frac{a}{1+a+b^{2}}+\frac{b}{1+b+c^{2}}+\frac{c}{1+c+a^{2}}\geq \frac{3}{4}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYmNjMmI1YmUtNWUzNC00ZWNkLThiODQtZWU4MzI0MjFkMWZi&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/chnx-sjyiL" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/chnx-sjyiL" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7195796586847449300?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-57371507210575527562010-03-03T23:48:00.002+07:002010-03-03T23:53:48.406+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?A,B,C" title="A,B,C" /> be three points of a circle of radius <img src="http://latex.codecogs.com/gif.latex?%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D," title="\dfrac{\sqrt{3}}{3}," /> and let $a,b,c$ be the side-lengths of triangle <img src="http://latex.codecogs.com/gif.latex?ABC." title="$ABC.$" /> Show that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%28b%5E2+c%5E2-a%5E2%29%28c%5E2+a%5E2-b%5E2%29%28a%5E2+b%5E2-c%5E2%29%20%5Cle%20a%5E4b%5E4c%5E4." title="(b^2+c^2-a^2)(c^2+a^2-b^2)(a^2+b^2-c^2) \le a^4b^4c^4." /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGU3MjkzYTctY2FiZC00MGE4LWJmMWQtMTFjOGUzYjcyYTU4&amp;hl=en">Download here.</a><br /><br /><a name='more'></a>&nbsp; </div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/JGiz_bxwu5" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/JGiz_bxwu5" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5737150721057552756?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-8007537738434583542010-03-02T08:45:00.002+07:002010-03-02T08:50:03.381+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7B1%7D%7Ba%5E2+bc%7D%5Cle%20%5Cfrac%7B%28a+b+c%29%5E2%7D%7B%5Csum%20a%5E2b%5E2+abc%5Csum%20a%7D.$$" title="$$\sum \frac{1}{a^2+bc}\le \frac{(a+b+c)^2}{\sum a^2b^2+abc\sum a}.$$" /></i></div><div style="text-align: justify;"><i> </i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNjg1MGU3YzQtNWU5OC00OTczLWFiNjUtYzZlMjkzOGFmMDMy&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/uhLLIwFIht" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/uhLLIwFIht" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-800753773843458354?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-17367922862595901222010-03-01T12:59:00.004+07:002010-03-01T13:11:47.921+07:00<div style="text-align: justify;"><i>Let $a,b,c,d$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><ul><li><i><img src="http://latex.codecogs.com/gif.latex?%28a%29%5C;%20%5Csum%20%5Cdfrac%7Ba%28b%5E%7B2%7D+2bc-4cd+da%29%7D%7B%28b+2c%29%5E%7B2%7D%7D%5Cgeq%200;" title="\sum \dfrac{a(b^{2}+2bc-4cd+da)}{(b+2c)^{2}}\geq 0;" /></i></li><li><i><img src="http://latex.codecogs.com/gif.latex?%28b%29%5C;%20%5Csum%20%5Cdfrac%7Bab%28d%5E%7B2%7D+2bc+ca-4bd%29%7D%7B%28a+2b%29%5E%7B2%7D%7D%5Cgeq%200." title="\sum \dfrac{ab(d^{2}+2bc+ca-4bd)}{(a+2b)^{2}}\geq 0." /> </i></li></ul><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZTNiMzM2YmUtZmRmMC00M2FjLThiODAtMDgzZDk2YzZiMjZi&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/wmtJEeA40w" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/wmtJEeA40w" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1736792286259590122?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-26701369616652444132010-03-01T11:59:00.003+07:002010-03-01T12:03:28.209+07:00<div style="text-align: justify;"><i>If <img src="http://latex.codecogs.com/gif.latex?a,b,c,k" title="a,b,c,k" /> are positive real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Csum%20%5Cfrac%7Bbc[b+c+%28k-1%29a]%7D%7Bka%5E%7B2%7D+bc%7D%5Cgeq%20a+b+c.$$" title="$$\sum \frac{bc[b+c+(k-1)a]}{ka^{2}+bc}\geq a+b+c.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNDFiZjIwNDAtZDY5MS00ZTI0LWFkZDktMjllMTgyZjkxM2E4&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/XNIzxoysL2" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/XNIzxoysL2" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2670136961665244413?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-84858236242000741612010-03-01T11:32:00.003+07:002010-03-01T11:40:01.062+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7Ba%7D%7Ba%5E%7B2%7D+2bc%7D+%5Cfrac%7Bb%7D%7Bb%5E%7B2%7D+2ca%7D+%5Cfrac%7Bc%7D%7Bc%5E%7B2%7D+2ab%7D%5Cleq%20%5Cfrac%7Ba+b+c%7D%7Bab+bc+ca%7D.$$" title="$$\frac{a}{a^{2}+2bc}+\frac{b}{b^{2}+2ca}+\frac{c}{c^{2}+2ab}\leq \frac{a+b+c}{ab+bc+ca}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNmY1ODg5NzItMjBkOC00M2ZiLWI3MzAtNTg1ZDI0YTdiYWE2&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/FoQxsasB0M" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/FoQxsasB0M" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8485823624200074161?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-31469187638603971932010-03-01T06:20:00.005+07:002010-03-01T06:25:38.069+07:00<div style="text-align: justify;"><i>If $a,b,c$ are real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Cbegin%7Balign*%7D%20%28a%5E%7B2%7D&amp;+2ab+3b%5E%7B2%7D%29%28b%5E%7B2%7D+2bc+3c%5E%7B2%7D%29%28c%5E%7B2%7D+2ca+3a%5E%7B2%7D%29%5Cge%20%5C%5C%20&amp;%5Cgeq%202[%28a%5E%7B2%7Db+b%5E%7B2%7Dc+c%5E%7B2%7Da%29+2%28ab%5E%7B2%7D+bc%5E%7B2%7D+ca%5E%7B2%7D%29+abc]%5E%7B2%7D.%20%5Cend%7Balign*%7D" title="\begin{align*} (a^{2}&amp;+2ab+3b^{2})(b^{2}+2bc+3c^{2})(c^{2}+2ca+3a^{2})\ge \\ &amp;\geq 2[(a^{2}b+b^{2}c+c^{2}a)+2(ab^{2}+bc^{2}+ca^{2})+abc]^{2}. \end{align*}" /></i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZTJmMGNhNDktM2U2OS00NTgxLWE4ZDAtOWZkODAwNDEwODIx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Sew36RFNe1" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Sew36RFNe1" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3146918763860397193?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-47739313905951417822010-02-27T20:56:00.002+07:002010-02-27T21:00:49.928+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c" title="a,b,c" /> be real numbers such that <img src="http://latex.codecogs.com/gif.latex?a+b+c=3." title="a+b+c=3." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?$$%5Cfrac%7B1%7D%7B2a%5E%7B2%7D+7%7D+%5Cfrac%7B1%7D%7B2b%5E%7B2%7D+7%7D+%5Cfrac%7B1%7D%7B2c%5E%7B2%7D+7%7D%5Cleq%20%5Cfrac%7B1%7D%7B3%7D.$$" title="$$\frac{1}{2a^{2}+7}+\frac{1}{2b^{2}+7}+\frac{1}{2c^{2}+7}\leq \frac{1}{3}.$$" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYTMzYzM2OTMtOTEyMi00ZDU1LWFjMzAtNDJiZDY2YzgyNDli&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/wZKwmIntyf" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/wZKwmIntyf" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4773931390595141782?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-382020459308383952010-02-26T18:39:00.003+07:002010-02-27T16:27:47.261+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a_1,a_2,%5Cldots,a_n" title="a_1,a_2,\ldots,a_n" /> be positive real numbers such that <img src="http://latex.codecogs.com/gif.latex?a_1+a_2+%5Ccdots+a_n=1." title="a_1+a_2+\cdots+a_n=1." /> Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Cbegin%7Barray%7D%7Bc%7D%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B1+a_%7B1%7D%7D+%5Cfrac%7B1%7D%7B1+a_%7B1%7D+a_%7B2%7D%7D+%5Ccdots%20+%5Cfrac%7B1%7D%7B1+a_%7B1%7D+a_%7B2%7D+%5Ccdots%20+a_%7Bn%7D%7D%3C%5C%5C%5C%5C%20%5Cdisplaystyle%3C%5Csqrt%7B%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%20%5Cfrac%7B1%7D%7Ba_%7B1%7D%7D+%5Cfrac%7B1%7D%7Ba_%7B2%7D%7D+%5Ccdots%20+%5Cfrac%7B1%7D%7Ba_%7Bn%7D%7D%5Cright%29%20%7D.%5Cend%7Barray%7D" title="\begin{array}{c}\displaystyle \frac{1}{1+a_{1}}+\frac{1}{1+a_{1}+a_{2}}+\cdots +\frac{1}{1+a_{1}+a_{2}+\cdots +a_{n}}&lt;\\\\ \displaystyle&lt;\sqrt{\frac{1}{2}\left( \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots +\frac{1}{a_{n}}\right) }.\end{array}" /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYWRlNTFiOWMtMzNlOC00NjdmLTgwZTktYjQyZGM0OWZlZTUx&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/u4AeG9VeWR" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/u4AeG9VeWR" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-38202045930838395?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-39151388282168275942010-02-26T08:23:00.001+07:002010-02-26T08:25:31.266+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c" title="a,b,c" /> be positive real numbers such that <img src="http://latex.codecogs.com/gif.latex?abc=1." title="abc=1." /> Prove that</i><br /><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Cfrac%7Ba%7D%7B%28a+3%29%5E2%7D+%5Cfrac%7Bb%7D%7B%28b+3%29%5E2%7D+%5Cfrac%7Bc%7D%7B%28c+3%29%5E2%7D%20%5Cle%20%5Cfrac%7B3%7D%7B16%7D." title="\frac{a}{(a+3)^2}+\frac{b}{(b+3)^2}+\frac{c}{(c+3)^2} \le \frac{3}{16}." /></i></div><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYTliNzMwZTYtN2UyOS00ZGI5LTg2NDItODdmMjc1MTcxMjRh&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/DXQ4stbq4O" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/DXQ4stbq4O" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3915138828216827594?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-30945145952777471252010-02-25T18:14:00.002+07:002010-02-25T18:21:45.669+07:00<div style="text-align: justify;"><i>Let <img src="http://latex.codecogs.com/gif.latex?a,b,c,d" title="a,b,c,d" /> be positive real numbers. If <img src="http://latex.codecogs.com/gif.latex?%5Cdfrac%7B1%7D%7B2%7D%5Cleq%20k%5Cleq%202," title="\dfrac{1}{2}\leq k\leq 2," /> then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Csum%5Cleft%28%20%5Cfrac%7Bka+b%7D%7Ba+b+c%7D%5Cright%29%20%5E%7B2%7D%5Cgeq%20%5Cfrac%7B4%7D%7B9%7D%28k+1%29%5E%7B2%7D." title="\sum\left( \frac{ka+b}{a+b+c}\right) ^{2}\geq \frac{4}{9}(k+1)^{2}." /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGY1MjE3ZGItNmM3My00NjRmLWE4MjAtZGE2YmFmMTY1NDk5&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/NJVICHpTDb" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/NJVICHpTDb" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3094514595277747125?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-14476063863648537692010-02-25T15:57:00.001+07:002010-02-25T15:58:31.830+07:00<div style="text-align: justify;"><i>&nbsp;Let <img src="http://latex.codecogs.com/gif.latex?a,b,c" title="a,b,c" /> be positive real numbers. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Csum%20a%5E%7B2%7Db%5E%7B2%7D%5Cleq%20%5Csum%20%5Cfrac%7Ba%5E%7B3%7D%28b%5E%7B3%7D+c%5E%7B3%7D%29%7D%7Ba%5E%7B2%7D+bc%7D%5Cleq%20%5Csum%20a%5E%7B4%7D." title="\sum a^{2}b^{2}\leq \sum \frac{a^{3}(b^{3}+c^{3})}{a^{2}+bc}\leq \sum a^{4}." /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXM2JkYjMyZmMtOGNmNS00ZDA1LTg4NjMtY2Q1NmM0OWIwMjM1&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/n-EOXQpK_D" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/n-EOXQpK_D" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1447606386364853769?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-62456973088637381652010-02-25T11:33:00.002+07:002010-02-25T11:36:34.117+07:00<div style="text-align: justify;"><i>If <img src="http://latex.codecogs.com/gif.latex?a,b,c" title="a,b,c" /> are the side-lengths of a triangle, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?a%5E%7B3%7D%28b+c%29+bc%28b%5E%7B2%7D+c%5E%7B2%7D%29%5Cgeq%20a%28b%5E%7B3%7D+c%5E%7B3%7D%29." title="a^{3}(b+c)+bc(b^{2}+c^{2})\geq a(b^{3}+c^{3})." /></i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZWQzYTRjZDMtNjA3YS00NjkxLWI1MTAtYjA4ZTdhNWU5ZjY2&amp;hl=en">Download here.</a><br /><i></i><br /><a name='more'></a><i><br /></i></div><div style="text-align: justify;"><i><br /></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/EVWzFzX1tr" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/EVWzFzX1tr" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6245697308863738165?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-11350445832784427672010-02-24T22:11:00.005+07:002010-02-24T22:21:02.485+07:00<div style="text-align: justify;"><i>If <img src="http://latex.codecogs.com/gif.latex?a,b,c,k" title="a,b,c,k" /> are real numbers, then</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Cbegin%7Barray%7D%7Bc%7D%28k-2%29[%20%28k+2%29%28%5Cdisplaystyle%20%5Csum_%7Bsym%7Da%5E%7B2%7Db+2%28k-1%29abc%29%20%5E%7B2%7D-%5C%5C%5Cbigskip%20-4%5Cdisplaystyle%5Cprod%20%28a%5E%7B2%7D+kab+b%5E%7B2%7D%29]%20%5Cgeq%200.%5Cend%7Barray%7D" title="\begin{array}{c}(k-2)[ (k+2)(\displaystyle \sum_{sym}a^{2}b+2(k-1)abc) ^{2}-\\\bigskip -4\displaystyle\prod (a^{2}+kab+b^{2})] \geq 0.\end{array}" /></i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZDA2MTRhNjgtN2ZlNi00NzBjLThkZmUtMDJkMmMzNjM1MzIx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/DX-G4lVB1n" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/DX-G4lVB1n" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1135044583278442767?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-18935547178651084942010-02-24T19:30:00.006+07:002010-02-25T11:11:50.352+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be the side-lengths of a triangle. Prove that if <img src="http://latex.codecogs.com/gif.latex?k%20%5Cge%208-2%5Csqrt%7B14%7D," /> then</i><br /><div style="text-align: center;"><i> <img src="http://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7Ba%5E%7B2%7D+kbc%7D,%5Cquad%5Cfrac%7B1%7D%7Bb%5E%7B2%7D+kca%7D,%5Cquad%5Cfrac%7B1%7D%7Bc%5E%7B2%7D+kab%7D" title="\frac{1}{a^{2}+kbc},\quad\frac{1}{b^{2}+kca},\quad\frac{1}{c^{2}+kab}" /></i></div><i> are also the side-lengths of a triangle.</i><br /><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNmU2ZTZjMGYtNDdkZS00YTMwLTgyM2UtOTM0ODU2YjIwYTdj&amp;hl=en">Download here.</a><br /><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/IFUHpWtbK3" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/IFUHpWtbK3" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1893554717865108494?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-4102973171276334232010-02-23T19:10:00.013+07:002010-02-24T17:39:00.530+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i><img src="http://latex.codecogs.com/gif.latex?%5Csum%20%5Cfrac%7Ba%28ab+bc-2ca%29%7D%7B%282a+b%29%5E2%7D%5Cge%200." title="\sum \frac{a(ab+bc-2ca)}{(2a+b)^2}\ge 0." /></i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><span style="font-weight: bold;"><span style="font-style: italic;"><i><b>Proof. </b></i></span></span><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZmJmYWZiODEtYjUzZS00ZmVmLWE1YjgtOWYyZWVkZDg4ZjE4&amp;hl=en">Download here.</a> </div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/6FVVOg9cCS" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/6FVVOg9cCS" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-410297317127633423?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-44830403537558316962010-02-21T10:19:00.002+07:002010-02-23T19:02:07.245+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be real numbers such that $a+b+c=3.$ Prove that</i></div><div style="text-align: center;"><i>$$\frac{|a+b|}{4a+b^{2}+c^{2}}+\frac{|b+c|}{4b+c^{2}+a^{2}}+\frac{|c+a|}{4c+a^{2}+b^{2}}\leq 1.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMTFkOWQyMWQtYjMxNC00ZTRkLWEzZjktMjU3N2YxYTY5NWVk&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/iYZPVtSc71" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/iYZPVtSc71" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4483040353755831696?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-43405486583419299762010-02-21T08:38:00.002+07:002010-02-23T18:47:59.139+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\geq 3\left( \frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}\right) ^{\frac{5}{12}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><span style="font-weight: bold;"><span style="font-style: italic;"><i><b>Proof. </b></i></span></span><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYWFkMGIxMDAtNWVkNC00MTlmLTkzZDMtOWVjOTJkZmJkNTVk&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/gR8RWWNm7t" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/gR8RWWNm7t" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4340548658341929976?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-67949068386163622002010-02-21T07:47:00.002+07:002010-02-23T18:49:32.437+07:00<div style="text-align: justify;"><i>Prove that for any real numbers $a,b,c,$ the inequality holds</i></div><div style="text-align: center;"><i>$$\sum\frac{a+b}{2a^{2}+3b^{2}+3c^{2}+8}\leq \frac{3}{8}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof.</b></i> <a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNWIwOTBjYzctMmYyZC00Y2Q4LWI1YjgtZTAyZTcxZDAwYWEz&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Y6-q9vlrAt" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Y6-q9vlrAt" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6794906838616362200?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-17140222559012122442010-02-19T17:24:00.002+07:002010-02-23T18:52:21.975+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers such that</i></div><div style="text-align: center;"><i>$$3(a+b+c) \ge 2\max\{a,b,c\}.$$</i></div><div style="text-align: justify;"><i>Prove that</i></div><div style="text-align: center;"><i>$$\sum \frac{a(b+c)}{a^2+bc}+\frac{25}{8}\cdot \frac{(a+b+c)^2}{a^2+b^2+c^2} \ge \frac{65}{8}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZTEyZjVjYWEtYTJkMy00ZjgzLWE4ZGItZTFlMjhkMDUwNjIy&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/KvrWjWQouS" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/KvrWjWQouS" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1714022255901212244?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-71504973001433821242010-02-19T10:57:00.002+07:002010-02-23T18:53:25.864+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$\sum \frac{ab(a+b)^{2}}{a^{2}+ab+b^{2}}\leq \frac{4\left( \sum a\right) \left( \sum a^{2}b^{2}+\sum a^{2}bc\right) }{3\sum ab(a+b)}.$$</i></div><div style="text-align: justify;"><i><br /></i></div><div style="text-align: justify;"><b><i>Proof. </i></b><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZTk4ZmFiNzMtZGMyYi00YmEzLThmOGYtNDlkM2Y5Mzk3Nzgx&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Ouq5xMiEcy" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Ouq5xMiEcy" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7150497300143382124?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-42377918232777688222010-02-17T11:23:00.002+07:002010-02-23T18:54:24.124+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be distinct real numbers such that $a+b+c=1$ and $ab+bc+ca&gt;0.$ Determine the minimum value of the following expression</i></div><div style="text-align: center;"><i>$$P(a,b,c)=\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXODYzY2ZkNDEtYTMyNi00MzA0LTk2ZjgtNWNjN2Q4MzY3NzI0&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Rvt8H8-zGl" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Rvt8H8-zGl" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4237791823277768822?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-60915702646918715352010-02-17T08:21:00.002+07:002010-02-23T18:55:34.483+07:00<div style="text-align: justify;"><i>If $a,b,c$ are the side-lengths of a triangle, then</i></div><div style="text-align: center;"><i>$$\frac{(b+c)^{2}}{a^{2}+2bc},\quad\frac{(c+a)^{2}}{b^{2}+2ca},\quad\frac{(a+b)^{2}}{c^{2}+2ab}$$</i></div><div style="text-align: justify;"><i>are also the side-lengths of a triangle.</i></div><div style="text-align: justify;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTZlMzRkYWItZDUxZC00OTUxLTgyY2EtMjMyYzE4OGQ4NDhh&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/VWoIdOp1Ka" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/VWoIdOp1Ka" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6091570264691871535?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-2684982515194077952010-02-16T19:51:00.003+07:002010-02-23T18:51:19.607+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$\frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b}\geq \frac{a(b+c)}{a^{2}+bc}+\frac{b(c+a)}{b^{2}+ca}+\frac{c(a+b)}{c^{2}+ab}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzQwOGY0OWEtODMxZi00ZjFlLWJiNjQtMDQxYTI5MWQ2OWZm&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/DzuuN9NCi0" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/DzuuN9NCi0" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-268498251519407795?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-72269706145319153402010-02-16T11:03:00.001+07:002010-02-23T18:56:34.146+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1.$ Prove that</i></div><div style="text-align: center;"><i>$$a+b+c+\frac{ab}{b+c}+\frac{bc}{c+a}+\frac{ca}{a+b} \ge \frac{3\sqrt{3}}{2}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYTY3OGM1YWItMWE3YS00NTI3LWJlODMtNjRjMGY5OWEwYTUw&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/ZpbJpax_pt" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/ZpbJpax_pt" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7226970614531915340?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-4743808210152549712010-02-16T00:20:00.004+07:002010-02-23T18:57:24.846+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\sum \frac{a^2}{b^2+c^2}+\frac{3}{2} \ge 2\sum \frac{bc}{a^2+bc}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZjU0MmNmYWEtMmI0ZC00NDlhLTg1MDQtZWJkOTdjNjA4ZDI3&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/t_Qa_aH19X" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/t_Qa_aH19X" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-474380821015254971?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-22126766531473994932010-02-15T23:26:00.002+07:002010-02-23T18:58:09.978+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\sum\frac{a(b+c)}{\sqrt{b^{2}+bc+c^{2}}}\geq 2\sqrt{ab+bc+ca}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNjQ2NDdkMGUtYTQ1ZC00OTg0LWJkMmYtYTA1MDhiOWI4ZjEz&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/H65IYEFUng" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/H65IYEFUng" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2212676653147399493?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-52513601815590522010-02-15T22:28:00.002+07:002010-02-23T18:59:13.278+07:00<div style="text-align: justify;"><i>If $a,b,c$ are the side-lengths of a triangle, then</i></div><div style="text-align: center;"><i>$$\sqrt{\frac{a(b+c)}{a^2+2bc}}, \quad \sqrt{\frac{b(c+a)}{b^2+2ca}},\quad \sqrt{\frac{c(a+b)}{c^2+2ab}}$$</i></div><div style="text-align: justify;"><i>are also the side-lengths of a triangle.</i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMDhjYmYyNzItNzVmNC00YWI3LWE4YzctN2JmNjNlOWUyMmEy&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/XnJ_MxZDZE" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/XnJ_MxZDZE" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5251360181559052?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-22754210393482864222010-02-15T10:38:00.002+07:002010-02-23T19:00:13.848+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\sum \sqrt{\frac{a^{2}+bc}{b+c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{a^{2}+b^{2}+c^{2}-ab-bc-ca}{\sqrt{2(a+b)(b+c)(c+a)}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYjJiNWUxMjMtMWZkMC00Nzg3LWJhMmItYTUxYWIyMWZlNjRk&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/zjP0Ve5wUc" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/zjP0Ve5wUc" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2275421039348286422?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-43287677810316852272010-02-15T09:50:00.001+07:002010-02-23T19:02:55.261+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\sum \frac{a}{b^2+bc+c^2} \ge \frac{2}{\sqrt{ab+bc+ca+\sqrt[3]{a^2b^2c^2}}}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMzBkOTZhYTUtNjQyOS00ODIzLTk1MjYtM2I2NDE0N2ZlYjUz&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/p040Agfhnt" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/p040Agfhnt" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4328767781031685227?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-40127924500264256292010-02-15T09:27:00.004+07:002010-02-23T19:04:55.263+07:00<div style="text-align: justify;"><i>If $a,b,c,d$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$abcd\prod (2a+b+c)\leq \prod (a+b)^{2}.$$</i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOGQzMTBmYTQtYmNmZC00MDQ4LWJkYmYtZDVlMDkzMTJkNTlm&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/blWgJcVUrv" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/blWgJcVUrv" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4012792450026425629?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-46105158088760707432010-02-14T22:46:00.001+07:002010-02-23T19:31:09.229+07:00<div style="text-align: justify;"><i>Let $a,$ $b,$ $c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\frac{a(b+c)}{a^{2}+2bc}+\frac{b(c+a)}{b^{2}+2ca}+\frac{c(a+b)}{c^{2}+2ab}\leq 1+\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><b><i>Proof. </i></b><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXY2Q1NmJmMDItYTFjMC00MzZiLTg3YjAtM2M3MTExZTc3NjE1&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/0p_R8hpAbY" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/0p_R8hpAbY" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4610515808876070743?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-56497731910984992932010-02-14T20:41:00.002+07:002010-02-23T19:32:08.692+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$\frac{2a^{2}+b^{2}}{c(a+b)}+\frac{2b^{2}+c^{2}}{a(b+c)}+\frac{2c^{2}+a^{2}}{b(c+a)}\geq \frac{9}{2}\cdot \frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzdhNzM3NWQtM2JlNi00ZjQ1LThkYmEtMGQ2ZmIyY2M2NmI5&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/8vyul4LTso" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/8vyul4LTso" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5649773191098499293?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-2618925437066872852010-02-14T19:58:00.002+07:002010-02-27T08:19:07.637+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be the side-lengths of a triangle. Prove that</i></div><div style="text-align: center;"><i>$$\frac{a(b+c)}{a^2+2bc}+\frac{b(c+a)}{b^2+2ca}+\frac{c(a+b)}{c^2+2ab} \le 2.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNjU4NzBjZGQtMTM5OC00YzM5LWI3MTctYjQxNzE0ZmFkMDJm&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/P2V-9Xnrjs" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/P2V-9Xnrjs" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-261892543706687285?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-72774479957133045822010-02-14T19:05:00.003+07:002010-02-23T19:34:06.270+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\frac{b+c}{2a^2+bc}+\frac{c+a}{2b^2+ca}+\frac{a+b}{2c^2+ab} \ge \frac{6}{a+b+c}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzBlMDQyODItN2I3YS00ZTA1LWE2NDctODY5OTY0MjZkYTgx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/QbcIbUlxYF" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/QbcIbUlxYF" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7277447995713304582?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-47010165951166144042010-02-14T17:06:00.002+07:002010-02-23T19:35:30.343+07:00<div style="text-align: justify;"><i>Let $a,$ $b,$ $c$ be distinct real numbers. Prove that for any real number $k,$ we have</i></div><div style="text-align: center;"><i>$$\left( \frac{a-kb}{a-b}\right) ^{2}+\left( \frac{b-kc}{b-c}\right)^{2}+\left( \frac{c-ka}{c-a}\right) ^{2}\geq k^{2}+1.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTk0NDBiNjktNTQ3ZC00NTQzLTg5YTctOTBlMDQ0MTA0YjNm&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/mBEj4KPCA1" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/mBEj4KPCA1" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4701016595116614404?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-24926215213368961382010-02-13T23:53:00.002+07:002010-02-23T19:36:29.865+07:00<div style="text-align: justify;"><i>If $a_{1}\geq a_{2}\geq \cdots \geq a_{n}\geq 0,$ then</i></div><div style="text-align: center;"><i>$$\sum_{i=1}^na_i-n\sqrt[n]{a_{1}a_{2}\cdots a_{n}}\geq \frac{1}{3}\left( \sqrt{a_{1}}+\sqrt{a_{2}}-2\sqrt{a_{n}}\right) ^{2}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZWZlMmVkMmQtYmViMy00ZTE1LWE2ZGEtNGM0OWY3YzMzMTlj&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/cymlsOIPrx" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/cymlsOIPrx" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2492621521336896138?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-14925079966770529312010-02-13T22:29:00.002+07:002010-02-23T19:37:09.431+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. If $k\geq 3+\sqrt{7},$ then</i></div><div style="text-align: center;"><i>$$\frac{a}{a^{2}+kbc}+\frac{b}{b^{2}+kca}+\frac{c}{c^{2}+kab}\geq \frac{9}{(k+1)(a+b+c)}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXODIzZmRhMWItNjY5ZC00Nzg0LThiNzQtNjljMTc4ZTkzNDg3&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/p-7CYqbAcv" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/p-7CYqbAcv" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1492507996677052931?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-29260482325121753122010-02-13T11:04:00.002+07:002010-02-23T19:38:22.182+07:00<div style="text-align: justify;"><i>If $a\geq b\geq c\geq d\geq 0,$ then</i></div><div style="text-align: center;"><i>$$a+b+c+d-4\sqrt[4]{abcd}\geq \frac{3}{2}\left( \sqrt{b}-2\sqrt{c}+\sqrt{d}\right) ^{2}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYmFhOGExZDQtZDllZi00NGYxLTgwOWUtNTBjODcyMTkzZDI4&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/h1qHIDaauI" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/h1qHIDaauI" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2926048232512175312?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-12483291550784249962010-02-13T02:28:00.003+07:002010-02-13T02:31:52.088+07:00<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/_ZZl4BL5SH3E/S3Wrh5kOT4I/AAAAAAAAAEY/QgpDzszAHZE/s1600-h/412MMMDHZZL.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/_ZZl4BL5SH3E/S3Wrh5kOT4I/AAAAAAAAAEY/QgpDzszAHZE/s320/412MMMDHZZL.jpg" /></a></div><div style="text-align: justify;"></div><div style="text-align: justify;"><b>Book title: </b>An Introduction to Inequalities.</div><div style="text-align: justify;"><b>Authors: </b>Edwin Beckenbach, Richard Bellman.<b><br /></b></div><div style="text-align: justify;"><b>Publisher: </b>The L. W. Singer Company.</div><div style="text-align: justify;"><b>ISBN:</b> 61-6228.</div><div style="text-align: justify;"><br /></div><div style="text-align: center;"><b>Preface</b></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement (appropriately by a philosopher) is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art. Secondly, the fundamental results of mathematics are often inequalities rather than equalities.</div><i> </i><br /><div style="text-align: justify;">In the pages that follow, we have presented three aspects of the theory of inequalities. First, in Chapters 1, 2, and 3, we have the axiomatic aspect. Secondly, in Chapter 4, we use the products of the preceding chapters to derive the basic inequalities of analysis, results that are used over and over by the practicing mathematician. In Chapter 5, we show how to use these results to derive a number of interesting and important maximum and minimum properties of the elementary symmetric figures of geometry: the square, cube, equilateral triangle, and so on. Finally, in Chapter 6, some properties of distance are studied and some unusual distance functions are exhibited.</div><div style="text-align: justify;"></div><i> </i><br /><div style="text-align: justify;">There is thus something for many tastes, material that may be read consecutively or separately. Some readers will want to understand the axiomatic approach that is basic to higher mathematics. They will enjoy the first three chapters. In addition, in Chapter 3 there are many illuminating graphs associated with inequalities. Other readers will prefer for the moment to take these results for granted and turn immediately to the more analytic results. They will find Chapter 4 to their taste. There will be some who are interested in the many ways in which the elementary inequalities can be used to solve problems that ordinarily are treated by means of calculus. Chapter 5 is intended for these. Readers interested in generalizing notions and results will enjoy the analysis of some strange non-Euclidean distances described in Chapter 6.</div><div style="text-align: justify;"></div><i> </i><br /><div style="text-align: justify;">Those whose appetites have been whetted by the material presented here will want to read the classic work on the subject, Inequalities, by G. H. Hardy, J. E. Littlewood, and G. P6lya, Cambridge University Press, London, 1934. A more recent work containing different types of results is Inequalities, by E. F. Beckenbach and R. Bellman, Ergebnisse der Mathematik, Julius Springer Verlag, Berlin, 1961.</div><i> </i><br /><div style="text-align: justify;"></div><div style="text-align: justify;"></div><div style="text-align: justify;"></div><div style="text-align: justify;"><a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/free%20public/An.Introduction.To.Inequalities%5E_Beckenbach.Bellman.djvu">Download</a></div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/ULSmj_p4Nd" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/ULSmj_p4Nd" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1248329155078424996?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-86393870491195172822010-02-12T23:01:00.006+07:002010-02-24T20:03:14.222+07:00<div style="text-align: justify;"><i>Let $a_{1}\leq a_{2}\leq \cdots \leq a_{n}&lt;b_{1}\leq b_{2}\leq \cdots \leq b_{n}&lt;c_{1}\leq c_{2}\leq \cdots$ $\leq c_{n}$ be real numbers. Prove that, for any positive real numbers $m_{1},$ $m_{2},\ldots ,m_{n},$ we have</i></div><div style="text-align: center;"><i>$$ \left[ \sum\limits_{i=1}^{n}m_{i}(a_{i}+b_{i}+c_{i})\right] ^{2}&gt;$$</i><br /><i>$$ &gt;\displaystyle 3\left(\sum\limits_{i=1}^{n}m_{i}\right) \left[\sum\limits_{i=1}^{n}m_{i}(a_{i}b_{i}+b_{i}c_{i}+c_{i}a_{i})\right] .$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNTYwM2RjZWUtYjNjMi00ZjFlLWFmMmUtMjlhN2MwM2RlNTFl&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/NB7bNtBoY0" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/NB7bNtBoY0" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8639387049119517282?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-35194906689698022892010-02-12T22:16:00.001+07:002010-02-24T20:04:03.210+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers such that $a+b+c=3,$ then</i></div><div style="text-align: center;"><i>$$\sqrt{3a+2b^3}+\sqrt{3b+2c^3}+\sqrt{3c+2a^3} \ge 3\sqrt{5}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTEyOTVkNGUtNjg5MS00Mjc5LWE4ZmEtMTNhYzdlMDFjOGQ3&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/f0VlldQyCb" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/f0VlldQyCb" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3519490668969802289?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-5812571827808985462010-02-12T20:44:00.001+07:002010-02-24T20:04:54.983+07:00<div style="text-align: justify;"><i>If $a,b,c$ are positive real numbers, then</i></div><div style="text-align: center;"><i>$$\frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b}+\frac{3(ab+bc+ca)}{(a+b+c)^2} \ge 4.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYmUxYzFjYmItMjY2MC00MzgxLTk2YWMtZTVmZDk5NTQ5ZjZj&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/fsHpfC3mFs" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/fsHpfC3mFs" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-581257182780898546?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-2205148883940467572010-02-12T09:56:00.002+07:002010-02-24T20:05:39.657+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\sum \sqrt{\frac{2a^2}{b+c}} \ge \sqrt{\frac{3(a^3+b^3+c^3)}{a^2+b^2+c^2}+2(a+b+c)}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXN2VmYWU2OGUtMTNjMi00YmYwLWIzODMtN2VlMTE5MjQ0MGRj&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/FUj51w8lGz" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/FUj51w8lGz" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-220514888394046757?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-27169752140787693682010-02-12T09:01:00.001+07:002010-02-24T20:06:31.659+07:00<div style="text-align: justify;"><i>If $a \ge b \ge c \ge d \ge 0,$ then</i></div><div style="text-align: center;"><i>$$a+b+c+d-4\sqrt[4]{abcd} \ge \frac{3}{8} \left( \sqrt{b}+\sqrt{c}-2\sqrt{d}\right)^2.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMzgyNWY2ZjMtOThmZS00ODRhLTg3ZjktM2VkNzJmNWFmZmQ4&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/adWxMOJgx9" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/adWxMOJgx9" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2716975214078769368?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-75571249435468770102010-02-11T16:49:00.002+07:002010-02-24T20:07:33.809+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i>$$2\sum \frac{a^2}{b+c} \ge \sum \sqrt{\frac{a^3+abc}{b+c}}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNTA0ZTQzZmUtMzY5Ny00OGZjLWFmN2QtMmFlNDM4MzMwMzJl&amp;hl=en">Download here.</a><br /><i></i><br /><a name='more'></a><i><br /></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Ma-wf8M7LF" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Ma-wf8M7LF" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7557124943546877010?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-66580337765798319162010-02-11T07:23:00.004+07:002010-02-24T20:08:24.774+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\frac{b+c-a}{5a^{2}+4bc}+\frac{c+a-b}{5b^{2}+4ca}+\frac{a+b-c}{5c^{2}+4ab}\geq \frac{1}{a+b+c}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><b><i>Proof. </i></b><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMjY4ZmIxMDgtZDE3YS00NTk2LTgzMTQtNjg1MTVkNTM2N2Vk&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/fsHpfC3mFs" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/fsHpfC3mFs" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6658033776579831916?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-81343557360654879572010-02-11T07:07:00.003+07:002010-02-24T20:09:58.699+07:00<div style="text-align: justify;"><i>If $a \ge b \ge c \ge d&gt;0,$ then</i></div><div style="text-align: center;"><i>$$a+b+c+d -4\sqrt[4]{abcd} \ge \frac{1}{2}\left( \sqrt{a}+\sqrt{b}-2\sqrt{c}\right)^2.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMWY3Yzc0N2EtZDc3OC00Zjk0LWFkOTQtZTM0ZTk1ZDVhMWNh&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/EfeqEFAUeB" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/EfeqEFAUeB" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-8134355736065487957?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-72579476581713129652010-02-10T22:27:00.004+07:002010-02-24T20:10:50.054+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be distinct real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\sum \left(\frac{a}{b-c}\right)^2+1 \ge \frac{3(a+b+c)^2}{4(a^2+b^2+c^2-ab-bc-ca)}.$$</i></div><br /><div style="text-align: justify;"><i><b>Proof.</b></i> <a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZGIzOTUxYzUtNWEzMC00NTRkLWJjY2ItNjQ3YTZkYWE4MjZk&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Zfa1yOJJ6J" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Zfa1yOJJ6J" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7257947658171312965?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-26915552133289854932010-02-10T19:36:00.002+07:002010-02-24T20:11:47.629+07:00<div style="text-align: justify;"><i>If $a,b,c$ are distinct real numbers, then</i></div><div style="text-align: center;"><i>$$\frac{(a-b)^2(a-c)^2}{(b-c)^4}+\frac{(b-c)^2(b-a)^2}{(c-a)^4}+\frac{(c-a)^2(c-b)^2}{(a-b)^4} \ge \frac{129}{16}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNmIxNjUyMDItOThkMi00NDI3LTk5NjItMDkwNjFmNTk4OTdl&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/HR36BB5tqy" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/HR36BB5tqy" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-2691555213328985493?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-31271097279842903962010-02-10T19:00:00.003+07:002010-02-24T20:12:31.809+07:00<div style="text-align: justify;"><i>If $a\geq b\geq c&gt;0,$ then</i></div><ul style="text-align: justify;"><li><i>(1) $a+b+c-3\sqrt[3]{abc}\geq \displaystyle \dfrac{64(a-b)^{2}}{7(11a+24b)};$</i></li><li><i>(2) $a+b+c-3\sqrt[3]{abc}\geq \displaystyle \dfrac{25(b-c)^{2}}{7(3b+11c)}.$</i></li></ul><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXN2VmMTg3NGEtYTQyMC00ZjA3LWE1ZTQtYmFkZGEyOWVmNDE2&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/FfxiJc_IlB" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/FfxiJc_IlB" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3127109727984290396?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-57989785487681318332010-02-09T23:10:00.004+07:002010-02-26T16:59:34.713+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be <b>real </b>numbers such that&nbsp;</i><br /><div style="text-align: center;"><i>$(a+b+c)(a^2+b^2)(b^2+c^2)(c^2+a^2) \ne0.$</i></div><i>Prove that</i></div><div style="text-align: center;"><i>$$\frac{4(a^2+b^2+c^2)}{(a+b+c)^2}+\sum\frac{ab}{a^2+ab+b^2}\ge \frac{7}{3}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMGEwMzM0MTItMTI5OC00YWIwLTg5MjUtYTAxNDYxNDEwMGY5&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/re-ygqh57Q" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/re-ygqh57Q" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5798978548768131833?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-13974193541988656972010-02-09T22:45:00.005+07:002010-02-24T20:13:32.369+07:00<div class="talk" style="text-align: justify;"><i>If $a,b,c,d$ are positive real numbers such that&nbsp;</i><br /><div style="text-align: center;"><i>$3(a^2+b^2+c^2+d^2)=4(a+b+c+d)+1,$</i></div><i> then</i></div><div class="talk" style="text-align: center;"><i>$$a^3+b^3+c^3+d^3 \le 11.$$</i></div><div class="talk" style="text-align: center;"><i><br /></i></div><div class="talk" style="text-align: justify;"><i><i><b>Proof. </b></i></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMjgzZWVhZTQtMjBlMC00NjAzLWJjNzgtNzNhNGVjYjI4MjA5&amp;hl=en">Download here.</a></div><div class="talk" style="text-align: justify;"></div><a name='more'></a><i><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/TfZLKH-l3j" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/TfZLKH-l3j" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object></i><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1397419354198865697?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-15594841485869346912010-02-09T22:38:00.002+07:002010-02-24T20:14:19.304+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be distinct real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\frac{(a-b)^2}{(b-c)^2}+\frac{(b-c)^2}{(c-a)^2}+\frac{(c-a)^2}{(a-b)^2} \ge 5.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzU5NzI4NzctNGZhMC00OTJlLWI3MDktMmI3ZTZkMTRlNDZi&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/C3v_OHfQXj" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/C3v_OHfQXj" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1559484148586934691?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-63248912577441737852010-02-09T22:34:00.003+07:002010-02-24T20:15:08.004+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be distinct real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\frac{(b-c)^4}{(a-b)^2(a-c)^2}+\frac{(c-a)^4}{(b-c)^2(b-a)^2}+\frac{(a-b)^4}{(c-a)^2(c-b)^2} \ge \frac{33}{2}.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMzAyYTIzNmItZTVjMi00YTljLTk3ODYtMTRlMWE5NzIyMDc2&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/r0C_4osd9O" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/r0C_4osd9O" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6324891257744173785?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-11660999779772671462010-02-06T00:02:00.004+07:002010-02-24T20:16:13.427+07:00<div style="text-align: justify;"><i>Let $a,b,c,d$ be nonnegative real numbers such that $a+b+c+d=4$ and $a^2+b^2+c^2+d^2=7.$ Prove that</i></div><div style="text-align: center;"><i>$$a^3+b^3+c^3+d^3 \le 16.$$</i></div><div style="text-align: center;"><i><br /></i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzJjYWU0YTItZTUyZS00OGYwLTg0NDYtZTE4OWVhYmFhMTY1&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/BzrAtZlP3c" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/BzrAtZlP3c" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1166099977977267146?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-79239450976114868382010-02-03T19:18:00.005+07:002010-02-24T20:17:01.913+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\sum\frac{a^5}{b^2+bc+c^2} \ge a^3+b^3+c^3-2abc.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOWNjOGFjMjAtMmVkNS00YmIxLThiY2YtM2VkY2UyZGRlYjcz&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/kExB_LHNBM" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/kExB_LHNBM" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7923945097611486838?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-8937258752191894292010-02-02T08:58:00.004+07:002010-02-24T20:18:16.922+07:00<div style="text-align: justify;"><i>If $a \ge b &gt;0,$ then</i></div><div style="text-align: center;"><i>$$a^{a-b} \ge 1-\frac{3(a-b)}{4\sqrt{a}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTE0MTQ5ODgtNGFkYS00ZDY1LTg1MjYtMTRjODMzMmY4ZjMy&amp;hl=en">Download here.</a><br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; <a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNDg1MTU3YzgtMmNkOC00YTM1LTk2ZmMtMTQ0MWUzODUwOGNl&amp;hl=en">Vasc's proof.</a> <br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/r73H90mbQ_" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/r73H90mbQ_" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-893725875219189429?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-7684860388544664012010-02-02T07:13:00.004+07:002010-02-24T20:19:13.942+07:00<div style="text-align: justify;"><i>If $a&gt;0$ and $b \ge 0$ such that $a \ge b,$ then</i></div><div style="text-align: center;"><i>$$a^{b-a} \le 1+\frac{a-b}{\sqrt{a}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzE1NWMwOTEtNmJlMC00ZjgwLWFiM2QtMWZkODllNWMwMGFm&amp;hl=en">Download here.</a><br />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; <a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNDg1MTU3YzgtMmNkOC00YTM1LTk2ZmMtMTQ0MWUzODUwOGNl&amp;hl=en">Vasc's proof.</a> </div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/YY7kR3mk32" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/YY7kR3mk32" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-768486038854466401?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-8903280649541881762010-02-01T20:31:00.001+07:002010-02-24T20:20:55.445+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be the side-lengths of a triangle. Prove that</i></div><div style="text-align: justify;"><i>$$\sqrt{9a^2+16bc}+\sqrt{9b^2+16ca}+\sqrt{9c^2+16ab} \ge 5(a+b+c).$$</i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYjlhODY3NjAtYjgyZC00NmVhLWJiZjgtOTRiNTYzMGQzYThk&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/lJlnABX4vV" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/lJlnABX4vV" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-890328064954188176?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-4563271254549053002010-01-31T16:41:00.002+07:002010-02-24T20:22:41.206+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers such that $a^2+b^2+c^2=1.$ Prove that</i></div><div style="text-align: center;"><i>$$\frac{1}{1-a^2}+\frac{1}{1-b^2}+\frac{1}{1-c^2}+\frac{1}{1-bc}+\frac{1}{1-ca}+\frac{1}{1-ab} \ge 9.$$</i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMjViYWVhMDItZDE4YS00YWU4LWJmOWUtYWNjMGUyZGQzNGU2&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/9DmF9AlLjf" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/9DmF9AlLjf" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-456327125454905300?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-51016355842044807942010-01-31T10:40:00.001+07:002010-02-24T20:23:32.827+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers such that</i></div><div style="text-align: center;"><i>$(4a+5b)(4b+5c)(4c+5a) =729.$</i></div><div style="text-align: justify;"><i>Prove that</i></div><div style="text-align: center;"><i>$abc(a^2+bc+ca)(b^2+ca+ab)(c^2+ab+bc) \le 27.$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMGM0MTE5MzgtMDY3Ni00YmQxLWExZjAtYjBjZWRkNjYyZjZj&amp;hl=en">Download here.</a></div><div style="text-align: justify;"></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/t9CLcIhdtp" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/t9CLcIhdtp" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5101635584204480794?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-2433313178190642042010-01-31T08:05:00.002+07:002010-02-24T20:24:52.849+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\frac{a(a^{2}+bc)}{b^{2}+bc+c^{2}}+\frac{b(b^{2}+ca)}{c^{2}+ca+a^{2}}+\frac{c(c^{2}+ab)}{a^{2}+ab+b^{2}}\geq \frac{2(a^{3}+b^{3}+c^{3})}{a^{2}+b^{2}+c^{2}}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXYzRlZDk3N2QtOTg4NS00NzUzLWFmNDUtZGY2NGI3NjM2YjI3&amp;hl=en">Download here.</a>&nbsp;</div><div style="text-align: justify;"><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/Dc-hsX8vgY" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/Dc-hsX8vgY" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-243331317819064204?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-8351663336310336152010-01-30T18:23:00.008+07:002010-02-24T20:25:36.709+07:00<div style="text-align: justify;"><i>Let $a,b,c,d$ be real numbers such that $a^2+b^2&lt;1$ and $c^2+d^2&lt;1.$ Prove that</i></div><div style="text-align: center;"><i>$$\frac{1-a}{1-a^2-b^2}+\frac{1-c}{1-c^2-d^2} \ge \frac{2-a-c}{1-\left(\displaystyle \frac{a+c}{2}\right)^2-\left(\frac{b+d}{2}\right)^2}.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMjMyNDVhYTktMGJjYS00NzYyLTliNjYtYTBkMWM4ZWI0ODA5&amp;hl=en">Download here.</a><br /><br /><a name='more'></a></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/oVi4NsTE2E" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/oVi4NsTE2E" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-835166333631033615?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-50418031195864454332010-01-27T09:24:00.001+07:002010-01-27T09:26:22.363+07:00<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/_ZZl4BL5SH3E/S1-hn0_aBjI/AAAAAAAAAEI/__l_DqAeFEY/s1600-h/stduviewerbox.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/_ZZl4BL5SH3E/S1-hn0_aBjI/AAAAAAAAAEI/__l_DqAeFEY/s320/stduviewerbox.gif" /></a><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i>Are you tired of many viewers one for each document?<br />Do you want to have, one program for all your technical documentations, scientific books and so on?<br />Do you want to have a program, which quickly opens files, quickly loads pages and quickly renders page to the screen?<br />And all of this in free software? </i><br /><br /></div><div style="text-align: justify;"><span style="font-size: large;"><b>View all most popular document formats using one program.</b></span><br /><br /><b>STDU Viewer</b> is a document viewer for multiple file formats. The goal of this software is to replace the multiple document viewers with a single simple application. <b>STDU Viewer</b> supports <b>TXT, TIFF, PDF, DjVu, Comic Book Archive (CBR or CBZ), XPS</b> document formats. These are the most popular formats for scientific and technical documentation. <br /><br /><span style="font-size: large;"><b>Use content for quick jumping from one section to another. </b></span><br /><br /><b>STDU Viewer</b> gives the possibility to show the content of documents for quick jumping from one section to another. <br /><br /><span style="font-size: large;"><b>Create your own bookmarks for quick access to a part of the document. </b></span><br /><br />You can create your own named bookmarks for easy access any part of the document. <br /><span style="font-size: large;"><b><br />STDU Viewer is free for non-commercial uses. </b></span><br /><br /><b>STDU Viewer</b> is provided as freeware for private non-commercial or educational use. <br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><span style="font-size: large;"><b>System Requirements </b></span><br /><ul><li><b>OS: </b>Microsoft Windows 2000/XP/2003/Vista/. </li><li><b>Processor: </b> Intel / AMD compatible at 1 GHz or higher. </li><li><b>RAM: </b>512 MB or higher. </li><li><b>Additional: </b><a href="http://www.microsoft.com/downloads/details.aspx?familyid=6A63AB9C-DF12-4D41-933C-BE590FEAA05A&amp;displaylang=en">GDI+</a> (exists on all system except Window 2000), Administrative permissions for program installation.</li></ul><a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Programs/stduviewer.exe">&nbsp;Download.</a><br /><br /><a name='more'></a>&nbsp; <br /></div><object width="1" height="1"><param name="movie" value="http://www.nhaccuatui.com/m/QCzEWIMVY8" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/QCzEWIMVY8" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-5041803119586445433?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-36892063627210496142010-01-27T08:51:00.005+07:002010-02-24T20:26:23.415+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers such that $a^2+b^2+c^2=1.$ Prove that</i></div><div style="text-align: center;"><i>$$7\sum \frac{1}{1-\left(\frac{b+c}{2}\right)^2}+3\sum \frac{1}{1-a^2} \ge 45.$$</i></div><div style="text-align: justify;"><br /><b><i>Proof. </i></b><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXMzE2NzFlMzUtNTNkNS00OGQ5LTk0N2MtNTBjM2QyNTNhNDNh&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><i></i><br /><a name='more'></a><i><br /></i></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/4nT_7tbf0X" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/4nT_7tbf0X" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-3689206362721049614?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-69947459095000908382010-01-25T17:53:00.001+07:002010-01-25T17:58:51.398+07:00<div style="text-align: justify;"><b>doPDF^7.1</b> installs itself as a virtual PDF printer driver so after a successful installation will appear in your <b>Printers and Faxes</b> list. To convert to PDF, you just have to print the document to doPDF, the free pdf converter. <b>Open</b> a document (with Microsoft Word, WordPad, NotePad or any other software), choose <b>Print</b> and select <b>doPDF</b>. It will ask you where to save the PDF file and when finished, the PDF file will be automatically opened in your default PDF viewer. Screenshot below shows how easy it is to <a href="http://www.dopdf.com/forum/topic/how-to-create-pdf-files-with-dopdf" title="create a PDF file">create a PDF file</a> from Microsoft Word (word to pdf): <br /></div><div style="padding-bottom: 5px; padding-top: 5px; text-align: center;"><img alt="Creating PDF files with doPDF" border="0" height="356" src="http://www.dopdf.com/images/free-pdf-converter.jpg" width="464" /><br /></div>Main benefits for using this free <b>PDF converter</b>: <br /><ul style="text-align: justify;"><li><b>It’s free</b> - PDF converter free for both commercial and personal use.</li><li><b>32-bit and 64-bit support</b> - can be installed on both 32-bit and 64-bit Windows operating systems.</li><li><b>No GhostScript or .NET</b> - doesn’t require 3rd party programs to do PDF files such as GhostScript or runtime environments like the .NET Framework. This makes the setup file to be incredibly small compared to other free PDF creator programs.</li><li><b>Customizable resolution</b> - you can choose any resolution from 72 to 2400 dpi.</li><li><b>Predefined/custom page sizes</b> - choose a default page size (Letter, Legal, A4, A5, A6, ...) or enter your own custom page size.</li><li><b>Searchable PDFs</b> - you can search for text within the created PDF file (and search engines will also index the text from the PDF).</li><li><b>Multi-language</b> - choose a language for the user interface from the 20 languages currently supported.</li><li><b>Do your part in saving the planet</b> - instead of consuming paper to print your document, why not create a PDF file out of it and spare a couple of planet-saving trees?</li><li><b>Barely uses any computer resources</b> - compared to other free PDF creator softwares, doPDF barely uses any memory or CPU resources when doing the actual conversion to PDF.</li></ul><div style="text-align: justify;">doPDF is a free PDF converter, it is not a PDF viewer. To view PDF files you will need the free Adobe Reader.<br /><br /><a href="http://cid-b2deb869e6bb1093.skydrive.live.com/self.aspx/Programs/dopdf.exe">Download. </a><br /><br /><a name='more'></a><br /><br /></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/lbwjKt6Rz7" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/lbwjKt6Rz7" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6994745909500090838?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-72170162106388510952010-01-25T00:23:00.001+07:002010-02-24T20:27:49.261+07:00<div style="text-align: justify;"><i>Let $0&lt;a,b \le 1.$ Prove that</i></div><div style="text-align: center;"><i>$a^{b-a}+b^{a-b} \le 2.$</i></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXZTgzNGI4N2ItZDkzYy00NzIwLTljOTAtODUyN2E5Y2ZiNjIz&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><br /></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/FoQxsasB0M" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/FoQxsasB0M" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7217016210638851095?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-62850916320834963662010-01-24T07:05:00.003+07:002010-02-24T20:28:38.620+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be nonnegative real numbers, no two of which are zero. Prove that</i></div><div style="text-align: center;"><i>$$\sum \frac{a^2(b+c)}{(b^2+c^2)(2a+b+c)} \ge \frac{2}{3}.$$</i></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXNThiNDA4YTMtYjI1NC00MjQwLWExNWItNjNkYzRiNzM3MTZj&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><br /></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/E6-oQKDgDR" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/E6-oQKDgDR" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-6285091632083496366?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-43357473130113468752010-01-22T11:05:00.001+07:002010-02-24T20:29:44.611+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers such that $a+b+c=ab+bc+ca.$ Prove that</i></div><div style="text-align: center;"><i>$$a^{\frac{a}{b}}b^{\frac{b}{c}}c^c \ge 1.$$</i></div><div style="text-align: center;"><br /></div><div style="text-align: justify;"><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOWUxMGFjYmMtZDg0ZC00NjQzLWI5YTYtOTliZmQ5NDE4N2Zl&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><br /></div><a name='more'></a><br /><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/vMNBMsS-PJ" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/vMNBMsS-PJ" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-4335747313011346875?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-11892931544368755112010-01-20T06:59:00.003+07:002010-02-24T20:31:12.125+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. If $k \ge 0,$ then</i></div><div style="text-align: center;"><i>$$\frac{1}{a\sqrt{a+kb}}+\frac{1}{b\sqrt{b+kc}}+\frac{1}{c\sqrt{c+ka}} \ge \frac{3}{\sqrt{(k+1)abc}}.$$</i></div><div style="text-align: justify;"><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXODNiMGU4YzktYTYzNC00NzUzLTkyMGYtMjA5YjZhNGYyMzRj&amp;hl=en">Download here.</a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><b>Remark.</b> For $k=1,$ we get the known inequality proposed by Phan Thanh Nam at VMEO 2005.<br /><br /><a name='more'></a><br /></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/dIgJloCM6i" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/dIgJloCM6i" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-1189293154436875511?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0tag:blogger.com,1999:blog-2044622312573518970.post-70417889724674523892010-01-20T05:33:00.006+07:002010-02-24T20:32:01.486+07:00<div style="text-align: justify;"><i>Let $a,b,c$ be positive real numbers. Prove that</i></div><div style="text-align: center;"><i>$$\sum \frac{a^4}{a^2+ab+b^2} \ge \frac{a^3+b^3+c^3}{a+b+c}.$$</i></div><div style="text-align: justify;"><br /><i><b>Proof. </b></i><a href="http://docs.google.com/leaf?id=0B6DvRg0tloIXOTUzNzE1ZDctN2Q4Ny00NDRmLTk2YTUtMzZjMDIwOWQ3NzAx&amp;hl=en">Download here.</a><br /><br /><a name='more'></a><br /></div><object height="1" width="1"><param name="movie" value="http://www.nhaccuatui.com/m/HOngGZb6DO" /><param name="quality" value="high" /><param name="wmode" value="transparent" /><param name="flashvars" value="&autostart=true" /><embed src="http://www.nhaccuatui.com/m/HOngGZb6DO" quality="high" wmode="transparent" type="application/x-shockwave-flash" flashvars="&autostart=true" width="1" height="1"></embed></object><div class="blogger-post-footer"><img width='1' height='1' src='https://blogger.googleusercontent.com/tracker/2044622312573518970-7041788972467452389?l=can-hang2007.blogspot.com' alt='' /></div>Võ Quốc Bá Cẩnhttp://www.blogger.com/profile/11426965121228851225noreply@blogger.com0