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

2005 Austrian-Polish Competition, 10
Tags: power , inequalities
Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$: \[1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.\]

2014 Thailand TSTST, 2
Tags: inequalities , geometric inequality , geometry
In a triangle $ABC$, let $x=\cos\frac{A-B}{2},y=\cos\frac{B-C}{2},z=\cos\frac{C-A}{2}$. Prove that $$x^4+y^4+z^2\leq 1+2x^2y^2z^2.$$

2025 China Team Selection Test, 18
Tags: inequalities , complex number
Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]

1974 Kurschak Competition, 3
Tags: inequalities , polynomial , algebra
Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$.

1991 National High School Mathematics League, 15
Tags: inequalities
If $0<a<1,x^2+y=0$, prove that $\log_a(a^x+a^y)\leq\log_a2+\frac{1}{8}$.

2007 Estonia Math Open Junior Contests, 5
Tags: inequalities , combinatorics
In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants m for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.

2023 Belarus Team Selection Test, 4.3
Tags: algebra , inequalities
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2010 Contests, 1
Tags: inequalities , number theory
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

2010 Postal Coaching, 5
Tags: inequalities , number theory
Prove that there exist a set of $2010$ natural numbers such that product of any $1006 $ numbers is divisible by product of remaining $1004$ numbers.

2011 Finnish National High School Mathematics Competition, 2
Tags: algebra , inequalities
Find all integers $x$ and $y$ satisfying the inequality \[x^4-12x^2+x^2y^2+30\leq 0.\]

2015 IMO Shortlist, A3
Tags: algebra , multivariate polynomial , maximization , n-variable inequality , inequalities
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2013 Math Prize For Girls Problems, 14
Tags: inequalities , ceiling function
How many positive integers $n$ satisfy the inequality \[ \left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ? \] Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

2019 Turkey EGMO TST, 2
Tags: algebra , inequalities , minimum value , three variable inequality
Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$. Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$

2004 IMC, 3
Tags: inequalities , triangle inequality
Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

1995 Tournament Of Towns, (452) 1
Tags: algebra , inequalities
Let $a, b, c$ and $d$ be points of the segment $[0,1]$ of the real line (this means numbers $x$ such that $0 \le x \le 1$). Prove that there exists a point $x$ on this segment such that $$\frac{1}{|x-a|}+\frac{1}{|x-b|}+\frac{1}{|x-c|}+\frac{1}{|x-d|}< 40.$$ (LD Kurliandchik)

2004 Bosnia and Herzegovina Team Selection Test, 3
Tags: algebra , inequality , positive real , inequalities
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$

2017 Caucasus Mathematical Olympiad, 6
Tags: inequalities
Given real numbers $a$, $b$, $c$ satisfy inequality $\left| \frac{a^2+b^2-c^2}{ab} \right|<2$. Prove that they also satisfy equalities $\left| \frac{b^2+c^2-a^2}{bc} \right|<2$ and $\left| \frac{c^2+a^2-b^2}{ca} \right| <2$.

2019 Centers of Excellency of Suceava, 1
Tags: inequalities , am-gm
Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $ [i]Gheorghe Stoica[/i]

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]

2011 Today's Calculation Of Integral, 696
Tags: algebra , inequalities , linear algebra , polynomial , calculus , integration , function
Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

2002 China Girls Math Olympiad, 5
Tags: floor function , inequalities
There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]

2014 AMC 10, 25
Tags: inequalities , floor function , logarithm , function
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

2007 Indonesia TST, 1
Tags: inequalities
Let $ a,b,c$ be real numbers. Prove that $ (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1)$.

2016 Mediterranean Mathematics Olympiad, 2
Tags: inequalities
Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that \[ \sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]

1960 Polish MO Finals, 1
Tags: inequalities , number theory , algebra
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.