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

2012 Online Math Open Problems, 7
Tags: probability , trigonometry , function , inequalities , number theory , relatively prime
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

2009 Sharygin Geometry Olympiad, 2
Tags: perimeter , inequalities , geometry
Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?

2013 ELMO Shortlist, 2
Tags: inequalities
Prove that for all positive reals $a,b,c$, \[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]

2011 Balkan MO Shortlist, A3
Tags: inequalities
Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

2009 China Team Selection Test, 3
Tags: inequalities
Let nonnegative real numbers $ a_{1},a_{2},a_{3},a_{4}$ satisfy $ a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1.$ Prove that $ max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2.$ Where for all integers $ i, a_{i \plus{} 4} \equal{} a_{i}$ holds.

2021 Ukraine National Mathematical Olympiad, 3
Tags: algebra , inequalities
For arbitrary positive reals $a\ge b \ge c$ prove the inequality: $$\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}$$ (Anton Trygub)

2019 Regional Olympiad of Mexico Southeast, 5
Tags: inequalities , set
Let $n$ a natural number and $A=\{1, 2, 3, \cdots, 2^{n+1}-1\}$. Prove that if we choose $2n+1$ elements differents of the set $A$, then among them are three distinct number $a,b$ and $c$ such that $$bc<2a^2<4bc$$

2015 Romania National Olympiad, 1
Tags: algebra , inequalities
Find all real numbers $x, y,z,t \in [0, \infty)$ so that $$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$

1999 India Regional Mathematical Olympiad, 5
Tags: inequalities
If $a,b,c$ are sides of a triangle, prove that \[ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3. \]

2018 Junior Balkan Team Selection Tests - Romania, 2
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{1}{a}+\frac{3}{b}+\frac{5}{c} \ge 4a^2 + 3b^2 + 2c^2$$ When does the equality hold? Marius Stanean

2010 Contests, 2
Tags: inequalities , trigonometry , calculus , derivative , algebra
Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

2008 239 Open Mathematical Olympiad, 2
Tags: inequalities , algebra
For all positive numbers $a, b, c$ satisfying $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$, prove that: $$ \frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .$$

2014 Saudi Arabia Pre-TST, 4.1
Tags: number theory , inequalities , divide , divisible
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

II Soros Olympiad 1995 - 96 (Russia), 9.5
Tags: inequalities , algebra
Solve the inequality $$3-2\left(3-2\left(3-...-2(3-2x)\right)...\right) >x$$. The total number of right parentheses is $100$.

2008 Baltic Way, 2
Tags: inequalities
Prove that if the real numbers $a,b$ and $c$ satisfy $a^2+b^2+c^2=3$ then \[\frac{a^2}{2+b+c^2}+\frac{b^2}{2+c+a^2}+\frac{c^2}{2+a+b^2}\ge\frac{(a+b+c)^2}{12}\] When does the inequality hold?

2025 Macedonian TST, Problem 5
Tags: geometry , inequalities , triangle inequality
Let $\triangle ABC$ be a triangle with side‐lengths $a,b,c$, incenter $I$, and circumradius $R$. Denote by $P$ the area of $\triangle ABC$, and let $P_1,\;P_2,\;P_3$ be the areas of triangles $\triangle ABI$, $\triangle BCI$, and $\triangle CAI$, respectively. Prove that \[ \frac{abc}{12R} \;\le\; \frac{P_1^2 + P_2^2 + P_3^2}{P} \;\le\; \frac{3R^3}{4\sqrt[3]{abc}}. \]

1999 National High School Mathematics League, 13
Tags: inequalities
If $x^2\cos\theta-x(1-x)+(1-x)^2\sin\theta>0$ for all $x\in[0,1]$, find the range value of $\theta$.

1984 IMO Longlists, 58
Tags: inequalities , limit , algebra
Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

2018 IFYM, Sozopol, 4
Tags: inequalities , algebra
$x \geq 0$ and $y$ are real numbers for which $y^2 \geq x(x + 1)$. Prove that: $(y - 1)^2 \geq x(x-1)$.

2007 CHKMO, 4
Tags: inequalities
Let a_1, a_2, a_3,... be a sequence of positive numbers. If there exists a positive number M such that for n = 1,2,3,..., $a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n}< Ma^{2}_{n+1}$ then prove that there exist a positive number M' such that for every n = 1,2,3,..., $a_{1}+a_{2}+...+a_{n}< M'a_{n+1}$

2017 Kyrgyzstan Regional Olympiad, 1
Tags: triangle inequality , inequalities
$a^3 + b^3 + 3abc \ge\ c^3$ prove that where a,b and c are sides of triangle.

2008 Singapore MO Open, 4
Tags: inequalities
let $0<a,b<\pi/2$. Show that $\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $

1967 Poland - Second Round, 1
Tags: algebra , inequalities
Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.

2018 CMIMC Algebra, 8
Tags: algebra , polynomial , inequalities
Suppose $P$ is a cubic polynomial satisfying $P(0) = 3$ and \[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\leq 0\] for all $x\in\mathbb R$. Determine all possible values of $P(-1)$.

2012 Mathcenter Contest + Longlist, 2 sl9
Tags: algebra , inequalities
Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of . $$a+b+c+\frac{3}{ab+bc+ca}$$ [i](PP-nine)[/i]