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

2021 Centroamerican and Caribbean Math Olympiad, 5
Tags: inequalities
Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that \[ a_1a_2 \cdots a_n \geq m^2M^{n-2} \] and determine when equality holds

2008 IMAC Arhimede, 2
Tags: geometry , geometric inequality , inequalities , angle bisector , circumcircle
In the $ ABC$ triangle, the bisector of $A $ intersects the $ [BC] $ at the point $ A_ {1} $ , and the circle circumscribed to the triangle $ ABC $ at the point $ A_ {2} $. Similarly are defined $ B_ {1} $ and $ B_ {2} $ , as well as $ C_ {1} $ and $ C_ {2} $. Prove that $$ \frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$

1985 Iran MO (2nd round), 3
Tags: inequalities , function , algebra
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that \[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\] Prove that \[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\] [i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]

JOM 2015 Shortlist, A1
Tags: inequality , inequalities
Let $ a, b, c $ be the side lengths of a triangle. Prove that $$ \displaystyle\sum_{cyc} \frac{(a^2 + b^2)(a + c)}{b} \ge 2(a^2 + b^2 + c^2) $$

2017 Romania National Olympiad, 4
Tags: algebra , inequalities
Let $a, b, c, d \in [0, 1]$. Prove that $$\frac{a}{1 + b}+\frac{b}{1 + c}+\frac{c}{1 + d}+\frac{d}{1 + a}+ abcd \le 3.$$

2016 Switzerland Team Selection Test, Problem 4
Tags: inequalities
Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied $$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$

2014 JBMO Shortlist, 8
Tags: algebra , inequalities , three variable inequality
Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$ if: (A) $\displaystyle {a = 0, b = 1}$ (B) $\displaystyle {a = 1, b = 0}$ (C) $\displaystyle {a + b = 1, \; a, b> 0}$ When the equality holds?

2012 Indonesia TST, 3
Tags: inequalities
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

2011 Mediterranean Mathematics Olympiad, 1
Tags: polynomial , vieta , inequalities , algebra
A Mediterranean polynomial has only real roots and it is of the form \[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial. [i](Proposed by Gerhard Woeginger, Austria)[/i]

2007 Moldova National Olympiad, 12.8
Tags: calculus , integration , inequalities , function , real analysis
Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

2011 Kurschak Competition, 3
Tags: combinatorial geometry , inequalities
Given $2n$ points and $3n$ lines on the plane. Prove that there is a point $P$ on the plane such that the sum of the distances of $P$ to the $3n$ lines is less than the sum of the distances of $P$ to the $2n$ points.

2025 Bulgarian Spring Mathematical Competition, 12.1
Tags: derivative , calculus , inequalities , algebra
In terms of the real numbers $a$ and $b$ determine the minimum value of $$ \sqrt{(x+a)^2+1}+\sqrt{(x+1-a)^2+1}+\sqrt{(x+b)^2+1}+\sqrt{(x+1-b)^2+1}$$ as well as all values of $x$ which attain it.

2000 Junior Balkan Team Selection Tests - Romania, 3
Tags: algebra , inequalities
Find all real numbers $ a $ such that $ x,y>a\implies x+y+xy>a. $ [i]Gheorghe Iurea[/i]

2010 Irish Math Olympiad, 3
Tags: inequalities
Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that (a) $xy+yz+xz\ge 9xyz$; (b) $xy+yz+xz<\frac{1}{4}+3xyz$;

1998 Chile National Olympiad, 4
Tags: algebra , inequalities
a) Prove that for any nonnegative real $x$, holds $$x^{\frac32} + 6x^{\frac54} + 8x^{\frac34}\ge 15x.$$ b) Determine all x for which the equality holds

2019 JBMO Shortlist, A6
Tags: inequalities , algebra
Let $a, b, c$ be positive real numbers. Prove the inequality $(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$. [i]Proposed by Tajikistan[/i]

2013 Korea National Olympiad, 2
Tags: inequalities , algebra
Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that \[ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }\]

2016 Switzerland - Final Round, 2
Tags: algebra , geometric inequalities , inequalities
Let $a, b$ and $c$ be the sides of a triangle, that is: $a + b > c$, $b + c > a$ and $c + a > b$. Show that: $$\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32$$

2000 Taiwan National Olympiad, 1
Tags: inequalities , number theory
Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

1988 Iran MO (2nd round), 2
Tags: inequalities , 3d geometry , tetrahedron , pythagorean theorem , geometry
In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that \[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]

2004 Gheorghe Vranceanu, 2
Tags: algebra , inequalities
[b]a)[/b] Let be an even number $ n\ge 4 $ and $ n $ positive real numbers $ x_1,x_2,\ldots ,x_n. $ Prove that: $$ \min_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}\le \frac{x_1+x_2+\cdots +x_{n/2}}{x_{1+n/2}+ x_{2+n/2} +\cdots + x_n}\le \max_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}$$ [b]b)[/b] Let be $ m\ge 1 $ pairwise distinct natural numbers $ a,b,\ldots ,c. $ Show that: $$ \frac{ab\cdots c}{a+b+\cdots +c}\ge (m-1)!\cdot\frac{2}{m+1} $$ [i]M. Tetiva[/i]

1998 Abels Math Contest (Norwegian MO), 1
Tags: sequence , inequalities , algebra
Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$. (a) Prove that $a_i < a_1^i$ for all $i > 1$. (b) Prove that $a_i > i$ for all $i$.

2010 IberoAmerican Olympiad For University Students, 5
Tags: linear algebra , matrix , algebra , polynomial , inequalities
Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

2010 India IMO Training Camp, 10
Tags: geometry , parallelogram , inequalities , analytic geometry , trigonometry , complex number
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

1999 Akdeniz University MO, 3
Tags: inequalities
For all $x> \sqrt 2$, $y> \sqrt 2$ numbers, prove that $$x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2$$