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

2010 Contests, 4
Tags: inequalities
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

2017 MMATHS, 1
Tags: algebra , number theory , inequalities
For any integer $n > 4$, prove that $2^n > n^2$.

2013 China Northern MO, 2
Tags: algebra , inequalities
If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.

2003 Brazil National Olympiad, 1
Tags: inequalities , triangle inequality , geometry , trigonometry
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

2005 Germany Team Selection Test, 3
Tags: inequalities , geometry
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

2002 Estonia Team Selection Test, 5
Tags: inequalities , trigonometry , algebra
Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

2014 India Regional Mathematical Olympiad, 4
Tags: least common multiple , vieta , symmetry , number theory , system of equations , inequalities , algebra
Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

1996 Spain Mathematical Olympiad, 3
Tags: algebra , triangle inequality , inequalities , function , polynomial
Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)|  \leq 2 $ for $ -1 \leq  x \leq 1 $.

1970 IMO Longlists, 15
Tags: perimeter , circumcircle , trigonometry , geometry , inequalities
Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$. Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.

2006 Croatia Team Selection Test, 4
Tags: number theory , inequalities
Find all natural solutions of $3^{x}= 2^{x}y+1.$

2016 Taiwan TST Round 2, 2
Tags: inequalities
Let $x,y$ be positive real numbers such that $x+y=1$. Prove that$\frac{x}{x^2+y^3}+\frac{y}{x^3+y^2}\leq2(\frac{x}{x+y^2}+\frac{y}{x^2+y})$.

2003 Kurschak Competition, 3
Tags: phi function , analytic number theory , expected value , probabilistic method , summation , number theory , inequalities
Prove that the following inequality holds with the exception of finitely many positive integers $n$: \[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]

the 11th XMO, 9
Tags: algebra , inequalities , trigonometry
$x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.$

2004 Kazakhstan National Olympiad, 1
Tags: inequalities
For reals $1\leq a\leq b \leq c \leq d \leq e \leq f$ prove inequality $(af + be + cd)(af + bd + ce) \leq (a + b^2 + c^3 )(d + e^2 + f^3 )$.

2014 Thailand TSTST, 2
Tags: inequalities
Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$

2013 Mexico National Olympiad, 6
Tags: inequalities , geometry
Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies \[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]

2017 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , logarithm , algebra , inequalities
In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers

2015 Balkan MO Shortlist, A2
Tags: circumradius , semiperimeter , geometric inequality , inequalities , inradius , geometry
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

1972 IMO Longlists, 9
Tags: inequalities
Given natural numbers $k$ and $n, k \le n, n \ge 3,$ find the set of all values in the interval $(0, \pi)$ that the $k^{th}-$largest among the interior angles of a convex $n$-gon can take.

2006 Iran MO (3rd Round), 1
Tags: inequalities
For positive numbers $x_{1},x_{2},\dots,x_{s}$, we know that $\prod_{i=1}^{s}x_{k}=1$. Prove that for each $m\geq n$ \[\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}\]

1974 Dutch Mathematical Olympiad, 3
Tags: algebra , number theory , inequalities
Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

2021 Malaysia IMONST 2, 2
Tags: inequalities
The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities $$a+b < c+d < e+a < b+c < d+e.$$ Among the five numbers, which one is the smallest, and which one is the largest?

2015 German National Olympiad, 6
Tags: mean , harmonic mean , inequalities
Prove that for all $x,y,z>0$, the inequality \[\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}\] holds. Determine if equality can hold and if so, in which cases it occurs.

2015 Indonesia MO Shortlist, A5
Tags: three variable inequality , algebra , inequalities
Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

2006 Junior Tuymaada Olympiad, 4
Tags: three variable inequality , algebra , inequalities
The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality $$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$