Found problems: 426
2010 China Team Selection Test, 2
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.
2000 Austrian-Polish Competition, 9
If three nonnegative reals $a$, $b$, $c$ satisfy $a+b+c=1$, prove that
$2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$.
1978 Canada National Olympiad, 3
Determine the largest real number $z$ such that
\begin{align*}
x + y + z = 5 \\
xy + yz + xz = 3
\end{align*}
and $x$, $y$ are also real.
2011 Turkey MO (2nd round), 3
$x,y,z$ positive real numbers such that $xyz=1$
Prove that:
$\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$
1986 China Team Selection Test, 3
Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that:
i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$
ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.
2006 China Girls Math Olympiad, 7
Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]
2008 Regional Competition For Advanced Students, 1
Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is:
\[ \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.\]
2012 Philippine MO, 3
If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]
2014 India Regional Mathematical Olympiad, 2
let $x,y$ be positive real numbers.
prove that
$ 4x^4+4y^3+5x^2+y+1\geq 12xy $
2010 China Team Selection Test, 1
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
1998 Greece National Olympiad, 3
Prove that for any non-zero real numbers $a, b, c,$
\[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]
2006 Hungary-Israel Binational, 2
If $ x$, $ y$, $ z$ are nonnegative real numbers with the sum $ 1$, find the maximum value of $ S \equal{} x^2(y \plus{} z) \plus{} y^2(z \plus{} x) \plus{} z^2(x \plus{} y)$ and $ C \equal{} x^2y \plus{} y^2z \plus{} z^2x$.
1972 IMO Longlists, 40
Prove the inequalities
\[\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}\]
2013 Moldova Team Selection Test, 1
For any positive real numbers $x,y,z$, prove that
$\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$
1978 IMO Longlists, 45
If $r > s >0$ and $a > b > c$, prove that
\[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]
VMEO II 2005, 8
If a,b,c>0, prove that:
\[ \frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}} \]
thank u for ur help :oops:
2009 China Team Selection Test, 3
Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$
2010 Tournament Of Towns, 5
For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.
2006 Iran MO (3rd Round), 5
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]
2014 South East Mathematical Olympiad, 4
Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]
2011 India IMO Training Camp, 1
Let $ABC$ be an acute-angled triangle. Let $AD,BE,CF$ be internal bisectors with $D, E, F$ on $BC, CA, AB$ respectively. Prove that
\[\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}\]
2012 Kazakhstan National Olympiad, 1
For a positive reals $ x_{1},...,x_{n} $ prove inequlity:
$ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$
2011 Czech-Polish-Slovak Match, 1
Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.
2003 Czech And Slovak Olympiad III A, 6
a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.
2011 Croatia Team Selection Test, 1
Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality
\[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]