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: 6530

1989 IMO Longlists, 19
Tags: inequalities , number theory
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds \[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]

2014 France Team Selection Test, 6
Tags: inequalities
Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

2009 Jozsef Wildt International Math Competition, W. 18
Tags: inequalities
If $a$, $b$, $c>0$ and $abc=1$, then $$\sum \limits^{cyc} \frac{a+b+c^n}{a^{2n+3}+b^{2n+3}+ab} \leq a^{n+1}+b^{n+1}+c^{n+1}$$ for all $n\in \mathbb{N}$

1980 Spain Mathematical Olympiad, 3
Tags: algebra , inequalities
Prove that if $a_1 , a_2 ,... , a_n$ are positive real numbers, then $$(a_1 + a_2 + ... + a_n) \left( \frac{1}{a_1}+ \frac{1}{a_1}+...+\frac{1}{a_n}\right)\ge n^2$$. When is equality valid?

1984 Canada National Olympiad, 5
Tags: trigonometry , algebra , inequalities
Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.

2001 Irish Math Olympiad, 4
Tags: inequalities
Prove that for all positive integers $ n$: $ \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}$.

1967 Bulgaria National Olympiad, Problem 2
Tags: algebra , inequalities , number theory
Prove that: (a) if $y<\frac12$ and $n\ge3$ is a natural number then $(y+1)^n\ge y^n+(1+2y)^\frac n2$; (b) if $x,y,z$ and $n\ge3$ are natural numbers for which $x^2-1\le2y$ then $x^n+y^n\ne z^n$.

2023 IMO, 4
Tags: inequalities
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that \[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\] is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

1961 Putnam, B4
Tags: absolute value , inequalities
Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$\sum_{i<j}|x_i -x_j |.$$

2019 Belarusian National Olympiad, 10.6
Tags: geometry , circumcircle , inequalities
The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively. Prove the inequality $AM+AL\ge 2AN$. [i](V. Karamzin)[/i]

2024 Thailand Mathematical Olympiad, 10
Tags: inequalities
Find the maximum value of \[abcd(a+b)(b+c)(c+d)(d+a)\] such that $a,b,c$ and $d$ are positive real numbers satisfying $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+\sqrt[3]{d}=4$

2015 Romania Team Selection Test, 5
Tags: algebra , maximization , sum , inequalities
Given an integer $N \geq 4$, determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.

2011 SEEMOUS, Problem 3
Tags: linear algebra , vector , inequalities
Given vectors $\overline a,\overline b,\overline c\in\mathbb R^n$, show that $$(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2$$where $\langle\overline x,\overline y\rangle$ denotes the scalar (inner) product of the vectors $\overline x$ and $\overline y$ and $\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle$.

1998 Baltic Way, 9
Tags: trigonometry , inequalities
Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions: $(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$; $(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$. Prove that $\gamma <\delta $.

2006 Moldova Team Selection Test, 3
Tags: inequalities
Positive real numbers $a,b,c$ satisfy the relation $abc=1$. Prove the inequality: $\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3$.

2020 Azerbaijan National Olympiad, 3
Tags: algebra , inequalities
$a,b,c$ are positive numbers.$a+b+c=3$ Prove that: $\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $

2017 Puerto Rico Team Selection Test, 5
Tags: max , min , algebra , maximum , inequalities , minimum
Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

2025 239 Open Mathematical Olympiad, 8
Tags: geometry , inequalities
The incircle of a right triangle $ABC$ touches its hypotenuse $BC$ at point $D$. The line $AD$ intersects the circumscribed circle at point $X$. Prove that $ |BX-CX| \geqslant |AD - DX|$.

1968 Miklós Schweitzer, 5
Tags: real analysis , inequalities
Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\] [i]P. Turan[/i]

2010 Junior Balkan Team Selection Tests - Moldova, 6
Tags: geometric inequalities , inequalities , geometry , algebra
In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.

2010 Indonesia TST, 3
Tags: algebra , induction , inequalities
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\] [i]Fajar Yuliawan, Bandung[/i]

1993 India National Olympiad, 3
Tags: invariant , inequalities , induction
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]

1998 AIME Problems, 2
Tags: perimeter , geometry , symmetry , parameterization , inequalities
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$

2012 South East Mathematical Olympiad, 2
Tags: inequalities
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.

2015 Stars Of Mathematics, 3
Tags: inequalities
Let $n$ be a positive integer and let $a_1,a_2,...,a_n$ be non-zero positive integers.Prove that $$\sum_{k=1}^n\frac{\sqrt{a_k}}{1+a_1+a_2+...+a_k}<\sum_{k=1}^{n^2}\frac{1}{k}.$$