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

2013 All-Russian Olympiad, 2
Tags: inequalities , inequalities proposed
Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

2002 Baltic Way, 4
Tags: inequalities , inequalities proposed
Let $n$ be a positive integer. Prove that \[\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 \] for all nonnegative real numbers $x_1,x_2,\ldots ,x_n$ such that $x_1+x_2+\ldots x_n=1$.

2014 ELMO Shortlist, 2
Tags: inequalities , inequalities proposed
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

2009 Indonesia TST, 2
Tags: inequalities , inequalities proposed
Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

2007 Macedonia National Olympiad, 1
Tags: inequalities , inequalities proposed
Let $a, b, c$ be positive real numbers. Prove that \[1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.\]

2010 Contests, 3
Tags: inequalities , inequalities proposed
Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

2008 Saint Petersburg Mathematical Olympiad, 7
Tags: inequalities , inequalities proposed
In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$. Fresh translation. This problem may be similar to one of the 9th grade problems.

2006 Irish Math Olympiad, 3
Tags: algebra , polynomial , inequalities proposed , inequalities
let x,y are positive and $ \in R$ that : $ x\plus{}2y\equal{}1$.prove that : \[ \frac{1}{x}\plus{}\frac{2}{y} \geq \frac{25}{1\plus{}48xy^2}\]

2012 Romania Team Selection Test, 1
Tags: inequalities , trigonometry , function , trig identities , Law of Cosines , inequalities proposed
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]

2007 Turkey Team Selection Test, 3
Tags: inequalities , function , inequalities proposed , algebra , highschoolmath
Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]

2002 Hungary-Israel Binational, 1
Tags: inequalities proposed , inequalities , algebra
Suppose that positive numbers $x$ and $y$ satisfy $x^{3}+y^{4}\leq x^{2}+y^{3}$. Prove that $x^{3}+y^{3}\leq 2.$

1996 Romania Team Selection Test, 14
Tags: inequalities , inequalities proposed
Let $ x,y,z $ be real numbers. Prove that the following conditions are equivalent: (i) $ x,y,z $ are positive numbers and $ \dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1 $; (ii) $ a^2x+b^2y+c^2z>d^2 $ holds for every quadrilateral with sides $ a,b,c,d $.

2001 Korea - Final Round, 3
Tags: inequalities , inequalities proposed
Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that \[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\] and find all cases of equality.

2012 ELMO Shortlist, 9
Tags: inequalities , algebra , polynomial , function , induction , inequalities proposed
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

2009 Czech and Slovak Olympiad III A, 3
Tags: inequalities , inequalities proposed
Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

Oliforum Contest II 2009, 4
Tags: inequalities , inequalities proposed
Let $ a,b,c$ be positive reals; show that $ \displaystyle a \plus{} b \plus{} c \leq \frac {bc}{b \plus{} c} \plus{} \frac {ca}{c \plus{} a} \plus{} \frac {ab}{a \plus{} b} \plus{} \frac {1}{2}\left(\frac {bc}{a} \plus{} \frac {ca}{b} \plus{} \frac {ab}{c}\right)$ [i](Darij Grinberg)[/i]

2023 All-Russian Olympiad Regional Round, 9.9
Tags: algebra , Russia , inequalities proposed , Inequality
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.

2007 Baltic Way, 4
Tags: inequalities proposed , inequalities
Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]

2005 Baltic Way, 5
Tags: inequalities , logarithms , inequalities proposed
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that \[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]

2007 China Northern MO, 2
Tags: inequalities , geometric inequality , inequalities proposed , China , High school olympiad
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]

2014 Romania National Olympiad, 1
Tags: inequalities , inequalities proposed
Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

2014 Moldova Team Selection Test, 2
Tags: inequalities , inequalities proposed
Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression: \[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]

1994 Irish Math Olympiad, 3
Tags: inequalities , inequalities proposed
Prove that for every integer $ n>1$, $ n((n\plus{}1)^{\frac{2}{n}}\minus{}1)<\displaystyle\sum_{i\equal{}1}^{n}\frac{2i\plus{}1}{i^2}<n(1\minus{}n^{\minus{}\frac{2}{n\minus{}1}})\plus{}4$.

2013 Saint Petersburg Mathematical Olympiad, 2
Tags: inequalities proposed , inequalities
if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \] A. Khrabrov

2010 Balkan MO Shortlist, A3
Tags: inequalities , inequalities unsolved , inequalities proposed , algebra
Let $a,b,c,d$ be positive real numbers. Prove that \[(\frac{a}{a+b})^{5}+(\frac{b}{b+c})^{5}+(\frac{c}{c+d})^{5}+(\frac{d}{d+a})^{5}\ge \frac{1}{8}\]