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

1973 Putnam, A3
Tags: floor function , inequalities , minimum
Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $$k+\frac{n}{k},$$ where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.$

2005 Federal Math Competition of S&M, Problem 3
Tags: inequalities , inequality
If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that $$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$

2006 France Team Selection Test, 2
Tags: algebra , inequalities
Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

2023 South East Mathematical Olympiad, 4
Tags: inequalities , number theory
Find the largest real number $c$, such that for any integer $s>1$, and positive integers $m, n$ coprime to $s$, we have$$ \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs$$ where $\{ x \} = x - \lfloor x \rfloor $.

2012 Balkan MO Shortlist, N3
Tags: function , inequalities , functional equation
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

2006 MOP Homework, 6
Tags: divisible , number theory , inequalities , subset
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

2006 Taiwan TST Round 1, 2
Tags: inequalities
Let $a_1<a_2<\cdots<a_n$ be positive integers. Prove that $\displaystyle a_n \ge \sqrt[3]{\frac{(a_1+a_2+\cdots+a_n)^2}{n}}$.

2006 Romania National Olympiad, 1
Tags: calculus , derivative , function , inequalities
Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$. [i]Dan Schwarz[/i]

1968 Dutch Mathematical Olympiad, 2
Tags: algebra , inequalities
It holds: $N,a > 0$. Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$, and if $N \ge 1$ and $a = [\sqrt{N}]$. Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$.

1999 China Team Selection Test, 1
Tags: inequalities , smoothing
For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.

Russian TST 2015, P1
Tags: algebra , inequalities
Let $n>4$ be a natural number. Prove that \[\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.\]

KoMaL A Problems 2022/2023, A.837
Tags: geometry , linear algebra , geometric inequality , inequalities , 3d geometry , tetrahedron
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that \[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \] [i]Submitted by Viktor Vígh, Szeged[/i]

2004 Baltic Way, 8
Tags: algebra , polynomial , inequalities , function , number theory , prime number , logarithm
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2010 District Olympiad, 2
Tags: inequalities
Consider two real numbers $ a\in [ - 2,\infty)\ ,\ r\in [0,\infty)$ and the natural number $ n\ge 1$. Show that: \[ r^{2n} + ar^n + 1\ge (1 - r)^{2n}\]

2022 ISI Entrance Examination, 8
Tags: trigonometry , algebra , inequalities
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.

1996 IMO Shortlist, 4
Tags: inequalities , polynomial , function , calculus , algebra
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

1975 Spain Mathematical Olympiad, 7
Tags: algebra , analytic geometry , inequalities
Consider the real function defined by $f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}$ for all $x \in R$. a) Determine its maximum. b) Graphic representation.

2017 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: parameterization , algebra , inequalities , logarithm
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

1998 China National Olympiad, 3
Tags: inequalities
Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$, satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$, find the maximal value of $|x_k|$.

2009 Mathcenter Contest, 1
Tags: inequalities , algebra
Let $m,n$ be natural numbers. Prove that $$m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}$$ [i](nooonuii)[/i]

2019 Jozsef Wildt International Math Competition, W. 54
Tags: inequalities , summation
Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$

2022 Korea Winter Program Practice Test, 2
Tags: polynomial , inequality , inequalities , algebra
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

2012 NIMO Problems, 10
Tags: inequalities
For reals $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \displaystyle \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$. [i]Proposed by Evan Chen[/i]

2019 Jozsef Wildt International Math Competition, W. 2
Tags: inequalities
If $0<a\leq c\leq b$ then $$\frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36b^{10}}\leq \frac{(b^{25}-a^{25})(b^{25}-c^{25})}{25}\leq \frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36(ac)^{10}}$$

2014 Moldova Team Selection Test, 2
Tags: inequalities
Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression: \[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]