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

2015 IFYM, Sozopol, 3
Tags: inequalities , inequalities proposed
Let $ a,b,c>0$ prove that:\[ \frac{a^{3}}{(a+b)^{3}}+\frac{b^{3}}{(b+c)^{3}}+\frac{c^{3}}{(c+a)^{3}}\geq \frac{3}{8} \] Good luck! :D

2005 Colombia Team Selection Test, 4
Tags: inequalities , induction , inequalities proposed
1. Prove the following inequality for positive reals $a_1,a_2...,a_n$ and $b_1,b_2...,b_n$: $(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})$

1976 IMO Longlists, 46
Tags: inequalities , symmetry , search , inequalities proposed
Let $ a,b,c,d$ be nonnegative real numbers. Prove that \[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]

2016 Iran Team Selection Test, 2
Tags: algebra , inequalities , inequalities proposed
Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$

2014 China Second Round Olympiad, 1
Tags: inequalities proposed , inequalities , China , BPSQ
Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]

1997 Iran MO (3rd Round), 4
Tags: inequalities , Cauchy Inequality , inequalities proposed
Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that \[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]

2014 Contests, 1
Tags: induction , inequalities , inequalities proposed
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.

2005 Kazakhstan National Olympiad, 2
Tags: inequalities , inequalities proposed , Kazakhstan
Prove that \[ab+bc+ca\ge 2(a+b+c)\] where $a,b,c$ are positive reals such that $a+b+c+2=abc$.

2014 ELMO Shortlist, 6
Tags: inequalities , inequalities proposed
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2012 Vietnam Team Selection Test, 2
Tags: inequalities , inequalities proposed
Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying: \[\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c \] then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.

2012 JBMO TST - Turkey, 3
Tags: inequalities , function , inequalities proposed
Show that for all real numbers $x, y$ satisfying $x+y \geq 0$ \[ (x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y) \]

2009 Poland - Second Round, 1
Tags: inequalities , induction , function , inequalities proposed
Let $a_1\ge a_2\ge \ldots \ge a_n>0$ be $n$ reals. Prove the inequality \[a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n\]

2007 Indonesia MO, 3
Tags: inequalities , inequalities proposed
Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca)$. Prove that these three inequalities hold: $ a\plus{}b>c$, $ b\plus{}c>a$, $ c\plus{}a>b$.

2013 China Second Round Olympiad, 3
Tags: inequalities proposed , inequalities
The integers $n>1$ is given . The positive integer $a_1,a_2,\cdots,a_n$ satisfing condition : (1) $a_1<a_2<\cdots<a_n$; (2) $\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}$ are all perfect squares . Prove that :$a_n\ge 2n^2-1.$

1992 Taiwan National Olympiad, 3
Tags: induction , inequalities proposed , inequalities , BPSQ , algebra , High school olympiad
If $x_{1},x_{2},...,x_{n}(n>2)$ are positive real numbers with $x_{1}+x_{2}+...+x_{n}=1$. Prove that $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.

2014 Contests, 3
Tags: inequalities proposed , inequalities
For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]

2011 South East Mathematical Olympiad, 2
Tags: inequalities , vector , inequalities proposed
Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .

2004 Junior Balkan Team Selection Tests - Romania, 2
Tags: inequalities , inequalities proposed
The real numbers $a_1,a_2,\ldots,a_{100}$ satisfy the relationship \[ a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101. \] Prove that $|a_k|\leq 10$, for all $k=1,2,\ldots,100$.

2015 Iran Team Selection Test, 6
Tags: inequalities , inequalities proposed , AM-GM
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that $$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$

2011 Turkey Team Selection Test, 1
Tags: inequalities , geometry , circumcircle , ratio , geometric transformation , homothety , inequalities proposed
Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that \[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]

2013 ELMO Shortlist, 8
Tags: inequalities , function , inequalities proposed
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

2017 Sharygin Geometry Olympiad, 2
Tags: inequalities proposed , geometry
If $ABC$ is acute triangle, prove distance from each vertex to corresponding excentre is less than sum of two greatest side of triangle

2013 Olympic Revenge, 3
Tags: inequalities , inequalities proposed
Let $a,b,c,d$ to be non negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that \[\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2\]

2010 Contests, 2
Tags: inequalities , inequalities proposed
Show that \[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \] for all positive real numbers $a, \: b, \: c.$

1990 Romania Team Selection Test, 2
Tags: inequalities , function , geometry , circumcircle , inequalities proposed
Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]