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.

AND:
OR:
NO:

Found problems: 6530

2005 South africa National Olympiad, 5
Tags: induction , n-variable inequality , inequalities
Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that \[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]

2007 Turkey MO (2nd round), 3
Tags: inequalities
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that $ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $

2011 Romania National Olympiad, 1
Tags: inequalities , algebra , factorization , sum of cubes , formula
Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

PEN O Problems, 36
Tags: induction , inequalities
Let a and b be non-negative integers such that $ab \ge c^{2}$ where $c$ is an integer. Prove that there is a positive integer n and integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$, $y_{1}$, $y_{2}$, $\cdots$, $y_{n}$ such that \[{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c\]

2016 Hanoi Open Mathematics Competitions, 3
Tags: inequalities , maximum , algebra
Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is (A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.

2012 Romania National Olympiad, 3
Tags: function , calculus , inequalities
[color=darkred]Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that: [list] [b]a)[/b] $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$ [b]b)[/b] $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$ [/list][/color]

2008 Kazakhstan National Olympiad, 1
Tags: inequalities , number theory
Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation: $ (2008\minus{}a_1)^2\plus{}(a_1\minus{}a_2)^2\plus{}\dots\plus{}(a_{2007}\minus{}a_{2008})^2\plus{}a_{2008}^2\equal{}2008$

2012 Romania National Olympiad, 4
Tags: function , integration , calculus , inequalities , ratio , real analysis
[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]

2018 Sharygin Geometry Olympiad, 3
Tags: inequalities , geometry , incenter
The vertices of a triangle $DEF$ lie on different sides of a triangle $ABC$. The lengths of the tangents from the incenter of $DEF$ to the excircles of $ABC$ are equal. Prove that $4S_{DEF} \ge S_{ABC}$. [i]Note: By $S_{XYZ}$ we denote the area of triangle $XYZ$.[/i]

2017 Swedish Mathematical Competition, 5
Tags: algebra , inequalities , geometric inequalities
Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle. (The maximal number of points is given for the best possible constant, with proof.)

1993 Romania Team Selection Test, 4
Tags: combination , combinatorics , binomial , max , inequalities
For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$

1991 IMTS, 4
Tags: geometry , 3d geometry , tetrahedron , inequalities
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that \[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]

2007 Princeton University Math Competition, 1
Tags: inequalities
Suppose that $A$ is a set of integers. Denote the number of elements in $A$ by $|A|$. Define $A+A = \{a_1+a_2: a_1, a_2 \in A\}$ and $A-A = \{a_1-a_2:a_1, a_2 \in A\}$. Prove or disprove: for any set $A$, we have the inequality $|A-A| \ge |A+A|$.

2011 Saudi Arabia Pre-TST, 1.4
Tags: fermat number , inequalities , algebra , number theory
Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

2010 Stars Of Mathematics, 3
Tags: inequalities
Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

2013 India National Olympiad, 6
Tags: algebra , polynomial , function , logarithm , inequalities
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

1999 Singapore Senior Math Olympiad, 3
Tags: increasing , decreasing , subsequence , algebra , inequalities , sequence
Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

2018 AIME Problems, 15
Tags: inequalities
Find the number of functions $f$ from $\{0,1,2,3,4,5,6\}$ to the integers such that $f(0)=0, f(6)=12$, and \[|x-y| \le |f(x)-f(y)| \le 3 |x-y| \]for all $x$ and $y$ in $\{0,1,2,3,4,5,6\}$.

2012 Flanders Math Olympiad, 3
Tags: trigonometry , inequalities , algebra
(a) Show that for any angle $\theta$ and for any natural number $m$: $$| \sin m\theta| \le m| \sin \theta|$$ (b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$: $$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$ (c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.

2005 Nordic, 2
Tags: calculus , algebra , inequalities
Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!) [b]EDITED with exponent 2 over c[/b]

1998 Argentina National Olympiad, 6
Tags: inequalities , algebra
Given $n$ non-negative real numbers, $n\geq 3$, such that the sum of the $n$ numbers is less than or equal to $3$ and the sum of the squares of the $n$ numbers is greater than or equal to $1$, prove that among the $n$ numbers three can be chosen whose sum is greater than or equal to $1$.

2012 AMC 10, 20
Tags: inequalities , number theory , modular arithmetic
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

2003 Baltic Way, 5
Tags: inequalities , induction , algebra
The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$ \[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]

Gheorghe Țițeica 2025, P2
Tags: inequalities
Let $a,b,c$ be three positive real numbers with $ab+bc+ca=4$. Find the minimum value of the expression $$E(a,b,c)=\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca}-(a-b)^2.$$

2005 Taiwan National Olympiad, 3
Tags: induction , function , inequalities
$a_1, a_2, ..., a_{95}$ are positive reals. Show that $\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$