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

1966 Polish MO Finals, 4
Tags: inequalities , algebra
ff nonnegative real numbers$ x_1,x_2,...,x_n$ satisfy $x_1 +...+x_n\le \frac12$, prove that $$(1-x_1)(1-x_2)...(1-x_n) \ge \frac12$$

1989 Cono Sur Olympiad, 3
Tags: probability , inequalities
Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.

2013 District Olympiad, 4
Tags: number theory , inequalities , algebra
For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

2004 Switzerland Team Selection Test, 4
Tags: inequalities
[i]Second Test, May 16[/i] Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that $\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ . When does equality hold?

1966 IMO Shortlist, 5
Tags: inequalities , trigonometry , geometric inequality
Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

1991 Romania Team Selection Test, 7
Tags: sum , algebra , inequalities
Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

2007 Bulgaria Team Selection Test, 2
Tags: inequalities , function , 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 $x,y\in(0,1].$

2003 Kazakhstan National Olympiad, 2
Tags: algebra , inequalities
For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$

2001 Moldova National Olympiad, Problem 2
Tags: inequality , optimization , inequalities
If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

2012 Romania Team Selection Test, 3
Tags: inequalities , vector , algebra
Let $m$ and $n$ be two positive integers for which $m<n$. $n$ distinct points $X_1,\ldots , X_n$ are in the interior of the unit disc and at least one of them is on its border. Prove that we can find $m$ distinct points $X_{i_1},\ldots , X_{i_m}$ so that the distance between their center of gravity and the center of the circle is at least $\frac{1}{1+2m(1- 1/n)}$.

1985 Polish MO Finals, 3
Tags: function , continuous , inequalities
The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.

2010 Indonesia TST, 4
Tags: inequalities , algebra
Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$ Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$

2022 Durer Math Competition Finals, 3
Tags: algebra , inequalities
Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

1979 IMO Longlists, 35
Tags: geometry , induction , area of a triangle , heron's formula , inequalities
Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

2025 Abelkonkurransen Finale, 4b
Tags: algebra , inequalities
Determine the largest real number \(C\) such that $$\frac{1}{x}+\frac{1}{2y}+\frac{1}{3z}\geqslant C$$ for all real numbers \(x,y,z\neq 0\) satisfying the equation $$\frac{x}{yz}+\frac{4y}{xz}+\frac{9z}{xy}=24$$

1995 IMC, 2
Tags: function , calculus , real analysis , integral , inequalities
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

1960 IMO, 2
Tags: inequalities , algebra
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

2017 Saudi Arabia IMO TST, 1
Tags: algebra , inequalities
Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that $$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$

2015 Saudi Arabia IMO TST, 3
Tags: inequalities , algebra , min , max
Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

2010 Balkan MO, 1
Tags: rearrangement inequality , inequalities
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

1978 Canada National Olympiad, 3
Tags: inequalities
Determine the largest real number $z$ such that \begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*} and $x$, $y$ are also real.

2022 Federal Competition For Advanced Students, P1, 1
Tags: inequalities , algebra
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality? [i](Walther Janous)[/i]

2014 CentroAmerican, 3
Tags: inequalities
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of \[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]

2012 VJIMC, Problem 4
Tags: inequalities
Let $a,b,c,x,y,z,t$ be positive real numbers with $1\le x,y,z\le4$. Prove that $$\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.$$

2005 Korea Junior Math Olympiad, 7
Tags: algebra , inequalities , inequality , n-variable inequality
If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$