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

1994 Czech And Slovak Olympiad IIIA, 6
Tags: inequalities , algebra
Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that $$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab} $$

1978 Austrian-Polish Competition, 3
Tags: trigonometry , inequalities
Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

2016 Tuymaada Olympiad, 4
Tags: inequalities
Non-negative numbers $a$, $b$, $c$ satisfy $a^2+b^2+c^2\geq 3$. Prove the inequality $$ (a+b+c)^3\geq 9(ab+bc+ca). $$

2025 Philippine MO, P5
Tags: inequalities , four variables , algebra
Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.

2020 IMO Shortlist, A3
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

1972 Dutch Mathematical Olympiad, 2
Tags: functional , inequalities , functional equation , functional inequality , algebra
Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and $\bullet$ $f(f(x)) = x$, for all $x \in R$.

2016 Turkmenistan Regional Math Olympiad, Problem 2
Tags: inequalities , geometry
If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$

2015 Romania National Olympiad, 2
Tags: algebra , inequalities
Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$

1983 Czech and Slovak Olympiad III A, 2
Tags: geometric inequality , inequality , geometry , inequalities
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.

1991 Romania Team Selection Test, 3
Tags: inequalities , coloring , combinatorics , polygon
Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$

1989 Greece National Olympiad, 3
Tags: algebra , polynomial , inequalities
If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.

2007 China Girls Math Olympiad, 6
Tags: inequalities
For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that $ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$

2012 Princeton University Math Competition, A2
Tags: algebra , inequalities
Let $a, b, c$ be real numbers such that $a+b+c=abc$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}$.

2003 Federal Math Competition of S&M, Problem 2
Tags: functional equation , functional inequality , inequalities
Let $ f : [0, 1] \to\ R $ be a function such that :- $1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ . $2.)$ $f(1) = 1$ . $3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ . Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.

1971 IMO Longlists, 54
Tags: inequalities , combinatorics
A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied: $(1)$ every member of the set $P$ knows every other member of the set $P$; $(2)$ no member of the set $P$ knows any other member of the set $P$.

2015 Saint Petersburg Mathematical Olympiad, 3
Tags: inequalities , geometry
$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$, Bisectors of angles $B$ and $C$ intersect in $L$. Prove $$2KL \geq |AB-BC+CD-DA|$$

2000 Greece National Olympiad, 3
Tags: inequalities
Find the maximum value of $k$ such that \[\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}\] holds for all positive numbers $x$ and $y.$

2016 Balkan MO Shortlist, A5
Tags: algebra , system of equations , inequalities , minimum , maximum
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

2007 Bulgarian Autumn Math Competition, Problem 12.3
Tags: symmetric inequality , three variable inequality , algebra , inequalities
Find all real numbers $r$, such that the inequality \[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\] holds for any real $a,b,c>0$.

2017 District Olympiad, 1
Tags: inequalities , limit , sequence
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $ [b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $ [b]b)[/b] Find the biggest real number $ a $ for which the following inequality is true: $$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$

2013 Tuymaada Olympiad, 4
Tags: inequalities , three variable inequality
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

1983 Polish MO Finals, 2
Tags: inequalities , algebra , irrational
Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$. Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.

2010 Contests, 525
Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

1992 Hungary-Israel Binational, 1
Tags: induction , function , inequalities
Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then \[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]

2007 Bulgaria Team Selection Test, 2
Tags: inequalities , number theory
Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$