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: 592

1971 IMO Longlists, 52
Tags: inequality , 4-variable inequality , four variables
Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

1967 IMO Shortlist, 3
Tags: trigonometry , trigonometric inequality , inequality , algebra
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

1994 IMO Shortlist, 2
Tags: number theory , inequality , algebra
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

2020 Junior Macedonian National Olympiad, 2
Tags: inequality , algebra
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

1967 IMO Longlists, 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 $x \in \left(0, \frac{\pi}{2} \right).$

1988 IMO Shortlist, 24
Tags: inequality system , inequality , sequence , linear recurrence
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

2013 Balkan MO Shortlist, A1
Tags: algebra , inequality
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$

2004 Croatia National Olympiad, Problem 1
Tags: complex number , inequality , complex inequality , inequalities
Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

2019 JBMO Shortlist, A1
Tags: algebra , inequality
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$. [i]Proposed by Serbia[/i]

2006 Petru Moroșan-Trident, 2
Tags: induction , inequalities , inequality
Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $ [b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then $$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$ for any natural numbers $ n. $ [b]b)[/b] Disprove the converse of [b]a).[/b] [i]Vasile Radu[/i]

2020 Junior Balkan Team Selection Tests-Serbia, 3#
Tags: inequality
Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.

2012 Polish MO Finals, 6
Tags: algebra , inequality , inequalities
Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.

2007 JBMO Shortlist, 2
Tags: algebra , inequality
$\boxed{\text{A2}}$ Prove that for all Positive reals $a,b,c$ $\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0$

2004 Mexico National Olympiad, 2
Tags: number theory , inequality , integer
Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$

2014 JBMO Shortlist, 7
Tags: three variable inequality , inequality
$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

2009 Philippine MO, 4
Tags: algebra , inequality , minimum value
Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

2022 Azerbaijan JBMO TST, A2
Tags: algebra , inequality
For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

1967 IMO Longlists, 37
Tags: inequality , three variable inequality , muirhead , algebra , inequalities
Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

2023 Mongolian Mathematical Olympiad, 1
Tags: algebra , inequality , inequalities
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]

2022 Grosman Mathematical Olympiad, P4
Tags: combinatorics , inequality
Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

Ukrainian TYM Qualifying - geometry, VII.12
Tags: geometry , geometric inequality , algebra , inequality
Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

1979 IMO Longlists, 29
Tags: inequality , maximization , calculus , optimization , lagrange
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

2019 Balkan MO, 2
Tags: inequality
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

2024 Turkey Junior National Olympiad, 4
Tags: algebra , inequality , inequalities
Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n>1$ be real numbers. Prove that the inequality below holds. $$\prod_{i=1}^n\left(a_ia_{i+1}-\frac{1}{a_ia_{i+1}}\right)\geq 2^n\prod_{i=1}^n\left(a_i-\frac{1}{a_i}\right)$$

2005 China Second Round Olympiad, 2
Tags: function , inequality , inequalities , algebra
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]