This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 787

1997 Irish Math Olympiad, 4
Tags: inequalities , inequalities proposed
Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that: $ a^2\plus{}b^2\plus{}c^2 \ge abc.$

2011 Canadian Students Math Olympiad, 3
Tags: inequalities proposed , inequalities
Find the largest $C \in \mathbb{R}$ such that \[\frac{x+z}{(x-z)^2} +\frac{x+w}{(x-w)^2} +\frac{y+z}{(y-z)^2}+\frac{y+w}{(y-w)^2} + \sum_{cyc} \frac{1}{x} \ge \frac{C}{x+y+z+w}\] where $x,y,z,w \in \mathbb{R^+}$. [i]Author: Hunter Spink[/i]

2002 Taiwan National Olympiad, 4
Tags: inequalities , inequalities proposed
Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.

2009 Petru Moroșan-Trident, 2
Tags: inequalities proposed , inequalities
If $ a,b,c>0$ and $ a\plus{}b\plus{}c\equal{}3$ prove that: $ \frac{{a^3 \left( {a \plus{} b} \right)}}{{a^2 \plus{} ab \plus{} b^2 }} \plus{} \frac{{b^3 \left( {b \plus{} c} \right)}}{{b^2 \plus{} bc \plus{} c^2 }} \plus{} \frac{{c^3 \left( {c \plus{} a} \right)}}{{c^2 \plus{} ac \plus{} a^2 }} \ge 2$ P.s. I dont know if it has been posted before.

2011 Vietnam Team Selection Test, 3
Tags: inequalities , inequalities proposed
Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: \[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\] Find the maximum and minimum value of the sum $S=x_1+x_2.$

2009 Indonesia TST, 1
Tags: inequalities , inequalities proposed
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 \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]

1997 Taiwan National Olympiad, 3
Tags: induction , inequalities proposed , inequalities
Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$(Here $a_{0}=a_{n},a_{n+1}=a_{1}$). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$.

2011 Balkan MO Shortlist, A3
Tags: inequalities , inequalities proposed , Balkan
Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

2011 Turkey Junior National Olympiad, 1
Tags: inequalities , inequalities proposed , algebra , Turkey
Show that \[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\] holds for all positive real numbers $x,y$.

2004 India IMO Training Camp, 3
Tags: inequalities , inequalities proposed
For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

2012 China Team Selection Test, 2
Tags: function , inequalities , vector , inequalities proposed
Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

2005 All-Russian Olympiad Regional Round, 10.2
Tags: inequalities , inequalities proposed
10.2 Prove for all $x>0$ and $n\in\mathbb{N}$ the following inequality \[1+x^{n+1}\geq \frac{(2x)^n}{(1+x)^{n-1}}.\] ([i]A. Khrabrov[/i])

2015 Saudi Arabia JBMO TST, 1
Tags: 3-variable inequality , inequalities , three variable inequality , algebraic inequality , Inequality , inequalities proposed
Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

2001 Romania National Olympiad, 1
Tags: inequalities proposed , inequalities
Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.

2008 IberoAmerican Olympiad For University Students, 3
Tags: inequalities , inequalities proposed
Prove that $x+\frac{1}{x^x}<2$ for $0<x<1$.

2012 APMO, 5
Tags: inequalities , LaTeX , function , inequalities proposed
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

1998 Vietnam National Olympiad, 2
Tags: analytic geometry , trigonometry , inequalities proposed , inequalities
Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.

2003 Switzerland Team Selection Test, 3
Tags: inequalities proposed , inequalities
Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

2012 Mediterranean Mathematics Olympiad, 2
Tags: inequalities , trigonometry , inequalities proposed
In an acute $\triangle ABC$, prove that \begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}

2003 CentroAmerican, 3
Tags: inequalities , calculus , logarithms , induction , inequalities proposed
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.

2012 China National Olympiad, 1
Tags: inequalities , inequalities proposed
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.

2000 Iran MO (3rd Round), 2
Tags: vector , inequalities proposed , inequalities
Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

2013 Macedonia National Olympiad, 4
Tags: inequalities proposed , inequalities
Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that \[ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 \]

1988 Romania Team Selection Test, 4
Tags: inequalities , induction , inequalities proposed
Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \] [i]Viorel Vajaitu[/i]

2008 Macedonia National Olympiad, 2
Tags: inequalities , function , inequalities proposed
Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a \plus{} b\right)\left(b \plus{} c\right)\left(c \plus{} a\right) \equal{} 8$. Prove the inequality \[ \frac {a \plus{} b \plus{} c}{3}\ge\sqrt [27]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3}} \]