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

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Found problems: 592

2008 Greece JBMO TST, 2
Tags: algebra , inequality
If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

1967 IMO Shortlist, 3
Tags: trigonometry , calculus , taylor series , inequality , trigonometric inequality
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

1994 Baltic Way, 14
Tags: geometry , inequality
Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality \[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

2021 Macedonian Mathematical Olympiad, Problem 1
Tags: sequence , recursive , inequality , factorial , am-gm , inequalities
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2017 239 Open Mathematical Olympiad, 7
Tags: inequality , inequalities
Find the greatest possible value of $s>0$, such that for any positive real numbers $a,b,c$, $$(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})^2 \geq s(\frac{1}{a^2+bc}+\frac{1}{b^2+ca}+\frac{1}{c^2+ab}).$$

2015 IMC, 10
Tags: inequality , polynomial
Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| &gt; \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

2015 Azerbaijan JBMO TST, 1
Tags: three variable inequality , inequality
$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

2018 Balkan MO Shortlist, A1
Tags: algebra , inequality
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2020 Baltic Way, 2
Tags: inequality , algebra
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$

2024 Brazil Team Selection Test, 3
Tags: algebra , inequality , induction
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

2020 Taiwan TST Round 1, 1
Tags: inequality , inequalities
Let $a$, $b$, $c$, $d$ be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*} Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}

2019 ELMO Shortlist, A1
Tags: algebra , inequality , inequalities
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

2014 India Regional Mathematical Olympiad, 2
Tags: inequalities , inequality , 3-variable inequality
Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.

2023 Tuymaada Olympiad, 1
Tags: algebra , inequality
Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$

2006 Federal Math Competition of S&M, Problem 1
Tags: inequality , inequalities , algebra
Let $x,y,z$ be positive numbers with the sum $1$. Prove that $$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$

1994 Korea National Olympiad, Problem 2
Tags: trigonometric inequality , trigonometry , inequality , geometry , algebra , inequalities
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that $csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$ and find the conditions for equality.

2022 Tuymaada Olympiad, 4
Tags: inequality , inequalities
For every positive $a_1, a_2, \dots, a_6$, prove the inequality \[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]

1975 IMO Shortlist, 12
Tags: trigonometry , trigonometric inequality , trigonometric identities , inequality , optimization
Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that \[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]

2013 JBMO Shortlist, 3
Tags: algebra , inequality
Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

2022 China Second Round, 3
Tags: algebra , inequality
Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that $(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$ and $a_i=0$ $(i>k);$ $(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$ $(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$ Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$

1987 Czech and Slovak Olympiad III A, 4
Tags: algebra , number theory , inequality , prime number , factorial , inequalities
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

1969 IMO Longlists, 35
Tags: inequality , series summation
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$

1992 Mexico National Olympiad, 5
Tags: inequalities , inequality , algebra
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$

2024 Sharygin Geometry Olympiad, 10.3
Tags: geo , geometry , inequality
Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.

2021 China Team Selection Test, 4
Tags: inequality , algebra
Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.