1/Prove that if x,y,x are real numbers,then
$ x^{3}(y^{2}+z^{2})^{2}+y^{3}(z^{2}+x^{2})^{2}+z^{3}(x^{2}+y^{2})^{2}\geq xyz[xy(x+y)^{2}+yz(y+z)^{2}+zx(z+x)^{2}]. $
USA TST 2009
2/Given real number x,y,z$ \in [\dfrac{1}2;2]$ and a,b,c are cyclic
$ \dfrac{60a^{2}-1}{4xy+5z}+\dfrac{60b^{2}-1}{4yz+5x}+\dfrac{60c^{2}-1}{4zx+5y}\geq 12 $
Modolva TST day 4
3/Let x,y,z ne non-negative real numbers.Prove that
$\[ \dfrac{x^{2}+y^{2}+z^{2}+xy+yz+zx}{6}\le\dfrac{x+y+z}{3}\cdot\sqrt{\dfrac{x^{2}+y^{2}+z^{2}}{3}} \]$
Hungary-Israel Binational » 2009
4/Let $a,b,c$ be positive real numbers. Prove that $\sqrt {\dfrac{{a^3 }}{{b^3 }}} + \sqrt {\dfrac{{b^3 }}{{c^3 }}} + \sqrt {\dfrac{{c^3 }}{{a^3 }}} \ge \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a}$ .
CẦN THƠ TST 2009
5/Pove that if $x,y,z$ are positive real numbers Let $M$is the muximun of x,y,z
$\ln z+\ln(\dfrac{x}{yz}+1),\ \ln\dfrac{1}{z}+\ln(xyz+1),\ \ln y+\ln(\dfrac{1}{xyz}+1)$Find the mniimun of $M$
ĐHSP TST 2009 (Vòng phụ)
6/ $a,b,c$ l be the side lengths of a traingle
Prove that
$\sum \sqrt{a}(\dfrac {1}{b+c-a}-\dfrac {1}{\sqrt{bc}})\geq 0$
7/Let be a,b,c be positive real number such that ab+bc+ca=3 Prove that:
$\[ \dfrac {1}{1+a^{2}(b+c)}+\dfrac {1}{1+b^{2}(c+a)}+\dfrac {1}{1+c^{2}(a+b) }\leq\dfrac {3}{1+2abc}. \]$
MathLinks Contest Edition 7 round 3
8/ Prove that if $a\geq b\geq c> 0$ such that abcd=1 , then
$\[ \dfrac {1}{1+a}+\dfrac {1}{1+b}+\dfrac {1}{1+c}\geq\dfrac{3}{1+\sqrt[3]{abc}}. \]$
MathLinks Contest Edition 7 round 7
9/Let a,b,c be positive real number such that a+b+c=3,Prove that
$ \dfrac{1}{2+a^{2}+b^{2}}+\dfrac{1}{2+b^{2}+c^{2}}+\dfrac{1}{2+c^{2}+a^{2}}\leq\dfrac{3}{4} $
Iran TST 2009
10/Let a,b,c be positive real number Prove that
$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\geq 0$
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e post lên đây có gì anh tổng hợp lại nhé
P/s:nếu có thể các bạn hãy post = tiếng anh nhé!_pm nếu mình dịch sai ^^
Edited by Janienguyen, 12-11-2009 - 17:19.
