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

2015 Korea National Olympiad, 3
Tags: inequalities
Reals $a,b,c,x,y$ satisfies $a^2+b^2+c^2+x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$

2005 Federal Math Competition of S&M, Problem 1
Tags: inequalities , inequality
If $x,y,z$ are positive numbers, prove that $$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$

2017 Azerbaijan Senior National Olympiad, A5
Tags: inequalities , inequality , algebra
$a,b,c \in (0,1)$ and $x,y,z \in ( 0, \infty)$ reals satisfies the condition $a^x=bc,b^y=ca,c^z=ab$. Prove that \[ \dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4} \] \\

1970 IMO Longlists, 50
Tags: inequalities , perimeter , area of a triangle , circumcircle , geometry
The area of a triangle is $S$ and the sum of the lengths of its sides is $L$. Prove that $36S \leq L^2\sqrt 3$ and give a necessary and sufficient condition for equality.

2010 Contests, 3
Tags: inequalities
Positive real $A$ is given. Find maximum value of $M$ for which inequality $ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $ holds for all $x, y>0$

MIPT Undergraduate Contest 2019, 2.2
Tags: calculus , inequalities , probability , integration , probability and stats , expectation , function
Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

2004 German National Olympiad, 5
Tags: inequalities
Prove that for four positive real numbers $a,b,c,d$ the following inequality holds and find all equality cases: $$a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.$$

2008 Bulgaria Team Selection Test, 2
Tags: geometry , trigonometry , inequalities , complex number
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

2020 Moldova Team Selection Test, 7
Tags: algebra , inequalities , inequality
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$

2015 Swedish Mathematical Competition, 3
Tags: inequalities , algebra , min
Let $a$, $b$, $c$ be positive real numbers. Determine the minimum value of the following expression $$ \frac{a^2+2b^2+4c^2}{b(a+2c)}$$

1966 IMO Shortlist, 13
Tags: inequalities , n-variable inequality
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality \[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]

2010 Abels Math Contest (Norwegian MO) Final, 2b
Tags: inequalities , algebra
Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

1986 Vietnam National Olympiad, 1
Tags: function , inequalities
Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]

2007 Nicolae Coculescu, 4
Tags: inequalities
Let be three nonnegative integers $ m,n,p $ and three real numbers $ x,y,z $ such that $ 2^mx+2^ny+2^pz\ge 0. $ Prove: $$ 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0 $$ [i]Cristinel Mortici[/i]

2007 Bulgaria Team Selection Test, 2
Tags: function , inequalities , algebra
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

2001 Moldova National Olympiad, Problem 8
Tags: inequalities
Suppose that $a,b,c$ are real numbers such that $\left|ax^2+bx+c\right|\le1$ for $-1\le x\le1$. Prove that $\left|cx^2+bx+a\right|\le2$ for $-1\le x\le1$.

2011 China Western Mathematical Olympiad, 2
Tags: trigonometry , inequalities
Let $a,b,c > 0$, prove that \[\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}\]

2012 China Second Round Olympiad, 3
Tags: inequalities
Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression \[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]

2002 All-Russian Olympiad, 4
Tags: inequalities , pigeonhole principle , number theory , floor function
From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.

2002 Hungary-Israel Binational, 2
Tags: inequalities , trigonometry , incenter , geometry
Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Define $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.

1995 All-Russian Olympiad, 1
Tags: inequalities , geometric sequence , number theory
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions? [i]A. Golovanov[/i]

2010 Contests, 2
Tags: inequalities
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that \[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]

2018 China Team Selection Test, 5
Tags: inequalities
Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have \[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda\] Where $a_{n+i}=a_i,i=1,2,\ldots,k$

2018 All-Russian Olympiad, 3
Tags: algebra , inequalities
Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that $$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$ (As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)

2006 MOP Homework, 3
Tags: number theory , greatest common divisor , inequalities
Prove that the following inequality holds with the exception of finitely many positive integers $n$: $\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.