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

2015 District Olympiad, 1
Tags: inequalities , induction , factorial
For any $ n\ge 2 $ natural, show that the following inequality holds: $$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$

1998 Greece Junior Math Olympiad, 2
Tags: inequalities
If $a_1, a_2,...., a_{n-1}, a_n$, are positive integers, prove that: $\frac{\prod_{i=1}^n(a_i^2+3a_i+1)}{a_1a_2....a_{n-1}a_n}\ge 2^{2n}$

2008 Austria Beginners' Competition, 3
Tags: inequalities , algebra
Prove the inequality $$\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}$$ for all real numbers $a$ and $b$ with $a + b\ne 0$. When does equality hold?

2022 VJIMC, 3
Tags: inequalities , probability theory
Let $x_1,\ldots,x_n$ be given real numbers with $0<m\le x_i\le M$ for each $i\in\{1,\ldots,n\}$. Let $X$ be the discrete random variable uniformly distributed on $\{x_1,\ldots,x_n\}$. The mean $\mu$ and the variance $\sigma^2$ of $X$ are defined as $$\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.$$ By $X^2$ denote the discrete random variable uniformly distributed on $\{x_1^2,\ldots,x_n^2\}$. Prove that $$\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).$$

2014 China Western Mathematical Olympiad, 6
Tags: inequalities , ceiling function
Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

2011 Kosovo National Mathematical Olympiad, 3
Tags: algebra , function , trigonometry , inequalities , calculus , system of equations
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

2002 Moldova Team Selection Test, 1
Tags: inequalities
Positive numbers $\alpha ,\beta , x_1, x_2,\ldots, x_n$ ($n \geq 1$) satisfy $x_1+x_2+\cdots+x_n = 1$. Prove that \[\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}.\] [b]Note.[/b] $x_{n+1}=x_1$.

1992 Baltic Way, 13
Tags: inequalities
Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality \[ \sum_{i\equal{}1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i\equal{}1}^n(x_i\plus{}y_i)^2} \] holds.

2007 Korea Junior Math Olympiad, 5
Tags: inequalities , inequality , three variable inequality , algebra
For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

2003 Tournament Of Towns, 2
Tags: inequalities , trigonometry , triangle inequality , geometry , circumcircle
Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

2024-IMOC, A1
Tags: algebra , inequalities , series
Given a positive integer $N$. Prove that \[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\] [i]Proposed by tan-1[/i]

1987 ITAMO, 5
Tags: algebra , sequence , inequalities
Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.

2011 India Regional Mathematical Olympiad, 6
Tags: inequalities
Find the largest real constant $\lambda$ such that \[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\] For all positive real numbers $a,b,c.$

1987 Tournament Of Towns, (147) 4
Tags: inequalities , radical , algebra
For any natural $n$ prove the inequality $$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$

2009 Kyiv Mathematical Festival, 5
Tags: sequence , recursion , recurrence relation , inequalities , algebra
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be? b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

2004 All-Russian Olympiad Regional Round, 9.6
Tags: algebra , inequalities
Positive numbers $x, y, z$ are such that the absolute value of the difference of any two of them are less than $2$. Prove that $$ \sqrt{xy +1}+\sqrt{yz + 1}+\sqrt{zx+ 1} > x+ y + z.$$

2009 Baltic Way, 2
Tags: polynomial , algebra , inequalities
Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality \[a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.\] Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.

1990 APMO, 2
Tags: algebra , polynomial , inequalities
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that \[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \] for $k = 1$, $2$, $\cdots$, $n - 1$.

2004 Moldova Team Selection Test, 4
Tags: function , inequalities , combinatorics
Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.

2014 ELMO Shortlist, 9
Tags: inequalities
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]

2010 Contests, 2a
Tags: algebra , inequalities
Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$

1994 Nordic, 1
Tags: inequalities , equilateral triangle
Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that $OA_1 + OB_1 + OC_1 < a$.

2016 Peru IMO TST, 1
Tags: algebra , inequalities
The positive real numbers $a, b, c$ with $abc = 1$ Show that: $\sqrt{a + \frac{1}{a}} + \sqrt{b + \frac{1}{b}} + \sqrt{c + \frac{1}{c}}\geq 2(\sqrt{a} + \sqrt{b} + \sqrt{c})$

2011 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: algebra , inequality , inequalities
If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$

2000 Macedonia National Olympiad, 2
Tags: inequalities
If $a_1,a_2,a_3\ldots a_n$ are positive numbers, find the maximum value of \[\frac{a_1a_2\ldots a_{n-1}a_n}{(1+a_1)(a_1+a_2)\ldots (a_{n-1}+a_n)(a_n+2^{n+1})} \]