《2010年数学奥林匹克不等式问题》上有 30 条评论

1. 問題1.(Albania BMO TST) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1818014&sid=9ae4875dcdd8df450f0f1e77759e5453#p1818014
   設$a,b,c$為某三角形的三邊長,對於不等式

   $$a^3+b^3+c^3<k(a+b+c)(ab+bc+ca)$$

   (a)證明$k=1$時不等式成立.
   (b)求使得不等式恒成立的最小常數$k$.

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2. 問題2.(Greece) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1790813&sid=9ae4875dcdd8df450f0f1e77759e5453#p1790813
   已知$x,y>0,x+y=2a$,求證:$x^3y^3(x^2+y^2)^2\le 4a^{10}.$

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3. 問題3. (India) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1882531&sid=9ae4875dcdd8df450f0f1e77759e5453#p1882531
   已知$a,b,c>0$且$ab+bc+ca\le 3abc$.求證:

   $$\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+\sqrt{\frac{a^2+b^2}{a+b}}+3\le \sqrt{2}(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}).$$

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4. 問題4. (India) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1883908&sid=9ae4875dcdd8df450f0f1e77759e5453#p1883908
   設$P$是三角形$ABC$的Brocard點,求證:

   $$\left(\frac{AP}{BC}\right)^2+\left(\frac{BP}{CA}\right)^2+\left(\frac{CP}{AB}\right)^2\ge 1.$$

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5. 問題5. (Iran) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1974763&sid=9ae4875dcdd8df450f0f1e77759e5453#p1974763
   已知$a,b,c>0$,求證:

   $$\frac{1}{a^2}+\frac{1}{b^2}+ \frac{1}{c^2}+ \frac{1}{(a+b+c)^2}\ge \frac{7}{25}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c}\right)^2.$$

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6. 問題6. (Iran) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1974770&sid=9ae4875dcdd8df450f0f1e77759e5453#p1974770
   已知$x,y,z>0,xy+yz+zx=1$,求證:

   $$3-\sqrt{3}+ \frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge (x+y+z)^2.$$

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7. 問題7. (Korea) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1826698&sid=9ae4875dcdd8df450f0f1e77759e5453#p1826698
   任意給定一個三角形$ABC,P,Q,R$是三角形$ABC$內切圓分別在邊 $BC,CA,AB$的切點.設$T$是三角形$ABC$的面積,$L$是其外接圓直徑,求證:

   $$\left(\frac{AB}{PQ}\right)^{3}+\left(\frac{BC}{QR}\right)^{3}+\left(\frac{CA}{RP}\right)^{3}\ge\frac{2}{\sqrt{3}}\frac{L^{2}}{T}.$$

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8. 問題8.(Portugal) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2000026&sid=9ae4875dcdd8df450f0f1e77759e5453#p2000026
   證明:對任意的三角形都存在某兩邊長$a,b$,滿足

   $$\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}.$$

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9. 問題9. (Vietnam) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2061442&sid=9ae4875dcdd8df450f0f1e77759e5453#p2061442
   已知$\displaystyle a,b,c>0,16(a+b+c)\ge \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.求證:

   $$\sum_{cyc}\left(\frac{1} {a+b+ \sqrt{2a+2c}}\right)^3\le \frac{8}{9}.$$

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10. 問題10. (Indonesia) http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1679875&sid=9ae4875dcdd8df450f0f1e77759e5453#p1679875
    已知$a,b,c\ge 0$和$x,y,z>0$且$a+b+c=x+y+z$,求證:

    $$\frac{a^3}{x^2}+\frac{b^3}{y^2}+\frac{c^3}{z^2}\ge a+b+c.$$

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