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

1967 German National Olympiad, 3
Tags: algebra , inequalities
Prove the following theorem: If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds $$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$

2000 Iran MO (3rd Round), 2
Tags: vector , inequalities
Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

2012 ELMO Shortlist, 6
Tags: inequalities , induction , calculus
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

2022 Junior Macedonian Mathematical Olympiad, P2
Tags: inequalities , algebra , rational function , function
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality $$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$ [i]Proposed by Anastasija Trajanova[/i]

2017 Caucasus Mathematical Olympiad, 6
Tags: inequalities
Given real numbers $a$, $b$, $c$ satisfy inequality $\left| \frac{a^2+b^2-c^2}{ab} \right|<2$. Prove that they also satisfy equalities $\left| \frac{b^2+c^2-a^2}{bc} \right|<2$ and $\left| \frac{c^2+a^2-b^2}{ca} \right| <2$.

2003 AIME Problems, 14
Tags: inequalities , number theory , relatively prime
The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.

2018 IFYM, Sozopol, 3
Tags: algebra , inequalities
The number 1 is a solution of the equation $(x + a)(x + b)(x + c)(x + d) = 16$, where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.

2018 Saudi Arabia JBMO TST, 2
Tags: inequalities
Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

2008 Romania National Olympiad, 2
Tags: inequalities
a) Prove that \[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n, \] for all positive integers $ n$. b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\} > 2^n$.

2008 China Team Selection Test, 2
Tags: inequalities
For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$

1995 Poland - First Round, 5
Tags: trigonometry , inequalities
Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that $(BP+CQ) \tan a/2 \geq BC$.

2002 IMC, 10
Tags: 3d geometry , tetrahedron , inequalities
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

2008 Peru MO (ONEM), 2
Tags: trigonometry , algebra , inequalities
Let $a$ and $b$ be real numbers for which the following is true: $acscx + b cot x \ge 1$, for all $0 <x < \pi$ Find the least value of $a^2 + b$.

2018 Czech-Polish-Slovak Match, 5
Tags: combinatorics , inequalities
In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]

2012 Mexico National Olympiad, 4
Tags: inequalities , combinatorics
The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$. How many integers less than $26000$ share the same [i]house[/i] as $2012$?

2021 Israel TST, 3
Tags: inequalities
What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

1999 All-Russian Olympiad Regional Round, 11.5
Tags: algebra , inequalities
Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds: $$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$

2014 Chile TST IMO, 1
Tags: inequalities , jensen
Given positive real numbers \(a\), \(b\), and \(c\) such that \(a+b+c \leq \frac{3}{2}\), find the minimum of \[ a+b+c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}. \]

2023 District Olympiad, P3
Tags: complex number , inequalities
Let $n\geqslant 2$ be an integer. Determine all complex numbers $z{}$ which satisfy \[|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.\]

2015 Silk Road, 1 (original)
Tags: inequalities , algebra
Given positive real numbers $a,b,c,d$ such that $ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$ Prove the inequality ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$

2013 Olympic Revenge, 3
Tags: inequalities
Let $a,b,c,d$ to be non negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that \[\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2\]

2014 ELMO Shortlist, 2
Tags: inequalities
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]

2016 Romania National Olympiad, 4
Tags: function , inequalities
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality $$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$ for all $a,b \in \mathbb R$.

2010 Contests, 3
Tags: polynomial , search , inequalities , algebra
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

1989 French Mathematical Olympiad, Problem 3
Tags: geometry , 3d geometry , tetrahedron , inequalities , geometric inequality , geometric inequalities
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.