Found problems: 787
2013 Bulgaria National Olympiad, 4
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$.
Prove that:
\[\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0\]
[i]Proposed by Nikolay Nikolov[/i]
2025 International Zhautykov Olympiad, 1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
2002 India IMO Training Camp, 5
Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that
\[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]
2009 Indonesia TST, 1
Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m\equal{}\min\{x_1,x_2,\dots,x_n\}$, $ M\equal{}\max\{x_1,x_2,\dots,x_n\}$, $ A\equal{}\frac{1}{n}(x_1\plus{}x_2\plus{}\dots\plus{}x_n)$, and $ G\equal{}\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]
2005 Croatia National Olympiad, 3
If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$
\[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]
2012 China Team Selection Test, 2
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
1992 IberoAmerican, 3
In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.
2005 Poland - Second Round, 3
Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then
\[\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2\]
2010 ELMO Shortlist, 6
For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\]
[i]In-Sung Na.[/i]
2014 China Northern MO, 6
Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.
2007 India IMO Training Camp, 2
Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that
\[a^2+b^2+c^2\leq 2abc+1.\]
2012 China Team Selection Test, 2
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
2005 International Zhautykov Olympiad, 1
For the positive real numbers $ a,b,c$ prove the inequality
\[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1.
\]
2010 Macedonia National Olympiad, 2
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
2013 Turkey MO (2nd round), 2
Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have
\[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]
1974 IMO Longlists, 50
Let $m$ and $n$ be natural numbers with $m>n$. Prove that
\[2(m-n)^2(m^2-n^2+1)\ge 2m^2-2mn+1\]
2012 Romania National Olympiad, 2
[color=darkred]Let $a$ , $b$ and $c$ be three complex numbers such that $a+b+c=0$ and $|a|=|b|=|c|=1$ . Prove that:
\[3\le |z-a|+|z-b|+|z-c|\le 4,\]
for any $z\in\mathbb{C}$ , $|z|\le 1\, .$[/color]
2024 Polish Junior MO Finals, 3
Real numbers $a,b,c$ satisfy $a+b \ne 0$, $b+c \ne 0$ and $c+a \ne 0$. Show that
\[\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.\]
2014 France Team Selection Test, 6
Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds:
\[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]
2016 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
2014 JBMO TST - Turkey, 4
Determine the smallest value of $(a+5)^2+(b-2)^2+(c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2+b^2+c^2-ab-bc-ca=3$.
2002 India IMO Training Camp, 20
Let $a,b,c$ be positive real numbers. Prove that
\[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]
1998 Iran MO (2nd round), 1
If $a_1<a_2<\cdots<a_n$ be real numbers, prove that:
\[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]
1995 Irish Math Olympiad, 1
Prove that for every positive integer $ n$,
$ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$
2007 Romania Team Selection Test, 2
Prove that for $n, p$ integers, $n \geq 4$ and $p \geq 4$, the proposition $\mathcal{P}(n, p)$
\[\sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}\quad \textrm{for}\quad x_{i}\in \mathbb{R}, \quad x_{i}> 0 , \quad i=1,\ldots,n \ ,\quad \sum_{i=1}^{n}x_{i}= n,\] is false.
[i]Dan Schwarz[/i]
[hide="Remark"]In the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions $\mathcal{P}(4, 3)$ are $\mathcal{P}(3, 4)$ true.[/hide]