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

2008 Ukraine Team Selection Test, 3
Tags: inequalities , inequalities proposed , 109
For positive $ a, b, c, d$ prove that $ (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)$

1990 IMO Longlists, 77
Tags: inequalities , inequalities proposed
Let $a, b, c \in \mathbb R$. Prove that \[(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.\] When does the equality hold?

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]

2012 Tuymaada Olympiad, 3
Tags: inequalities , inequalities proposed
Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds \[\dfrac{1} {2a^2+b^2+3}+\dfrac {1} {2b^2+c^2+3}+\dfrac{1} {2c^2+a^2+3}\leq \dfrac {1} {2}.\] [i]Proposed by V. Aksenov[/i]

2015 China National Olympiad, 1
Tags: inequalities , trigonometry , complex numbers , inequalities proposed
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that \[ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).\]

2008 Bulgaria Team Selection Test, 2
Tags: inequalities , trigonometry , complex numbers , inequalities proposed , geometry
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

2025 Abelkonkurransen Finale, 4b
Tags: algebra , inequalities proposed
Determine the largest real number \(C\) such that $$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$ for all real numbers \(x,y,z\neq 0\) satisfying the equation $$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$

2002 Mediterranean Mathematics Olympiad, 4
Tags: vector , inequalities proposed , inequalities
If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that: \[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]

2009 Moldova Team Selection Test, 4
Tags: inequalities , inequalities proposed
let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality: $ \dfrac{60a^2\minus{}1}{4xy\plus{}5z}\plus{}\dfrac{60b^2\minus{}1}{4yz\plus{}5x}\plus{}\dfrac{60c^2\minus{}1}{4zx\plus{}5y}\geq 12$

2008 China Team Selection Test, 2
Tags: inequalities , algebra , polynomial , inequalities proposed
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

2010 Korea National Olympiad, 2
Tags: inequalities , inequalities proposed
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that \[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]

2012 Romania Team Selection Test, 2
Tags: inequalities , function , inequalities proposed
Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]

2010 China Second Round Olympiad, 3
Tags: inequalities proposed , inequalities
let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

2012 China Girls Math Olympiad, 1
Tags: inequalities , induction , inequalities proposed , Hi , marvio
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that $\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

2013 Saint Petersburg Mathematical Olympiad, 5
Tags: inequalities , induction , inequalities proposed
Let $x_1$, ... , $x_{n+1} \in [0,1] $ and $x_1=x_{n+1} $. Prove that \[ \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1. \] A. Khrabrov, F. Petrov

2012 Turkey Junior National Olympiad, 3
Tags: inequalities , inequalities proposed
Let $a, b, c$ be positive real numbers satisfying $a^3+b^3+c^3=a^4+b^4+c^4$. Show that \[ \frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1 \]

2008 Junior Balkan Team Selection Tests - Romania, 4
Tags: inequalities proposed , inequalities
Determine the maximum possible real value of the number $ k$, such that \[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\] for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$.

2002 China National Olympiad, 3
Tags: inequalities proposed , inequalities
Suppose that $c\in\left(\frac{1}{2},1\right)$. Find the least $M$ such that for every integer $n\ge 2$ and real numbers $0<a_1\le a_2\le\ldots \le a_n$, if $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$, then we always have that $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$ where $m=[cn]$

2008 Hong Kong TST, 2
Tags: limit , inequalities proposed , inequalities
Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]

2001 China Western Mathematical Olympiad, 4
Tags: inequalities , inequalities unsolved , inequalities proposed
Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.

2003 Baltic Way, 4
Tags: inequalities , inequalities proposed
Let $a,b,c$ be positive real numbers. Prove that \[ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]

2015 China Team Selection Test, 2
Tags: inequalities , inequalities proposed , China TST
Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]

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]

1986 IMO Longlists, 1
Tags: inequalities , inequalities proposed
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality \[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\] for all real $b \geq 0.$

2014 Tajikistan Team Selection Test, 3
Tags: inequalities , geometry , inequalities proposed
Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality \begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}