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

2020 Germany Team Selection Test, 1
Tags: algebra , inequalities
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2015 China Team Selection Test, 2
Tags: number theory , inequalities
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.

2014 Belarus Team Selection Test, 2
Tags: algebra , inequalities
Prove that for all even positive integers $n$ the following inequality holds a) $\{n\sqrt6\} > \frac{1}{n}$ b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $ (I. Voronovich)

2018 Ramnicean Hope, 1
Tags: geometry , trigonometry , algebra , inequalities
Show that $ 2/3+\sin 2018^{\circ } >0. $ [i]Costică Ambrinoc[/i]

2010 JBMO Shortlist, 5
Tags: inequalities
Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $ Prove that $xyz \leq 36$

2004 Turkey Junior National Olympiad, 3
Tags: inequalities
On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?

1968 Yugoslav Team Selection Test, Problem 1
Tags: inequalities , combinatorics , geometry
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

2011 National Olympiad First Round, 19
Tags: inequalities
For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points? $\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$

2003 Abels Math Contest (Norwegian MO), 1b
Tags: sum , inequalities , algebra
Let $x_1,x_2,...,x_n$ be real numbers in an interval $[m,M]$ such that $\sum_{i=1}^n x_i = 0$. Show that $\sum_{i=1}^n x_i ^2 \le -nmM$

2010 Germany Team Selection Test, 2
Tags: inequalities
Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

2019 Saudi Arabia JBMO TST, 4
Tags: inequalities
Let $a, b, c$ be positive reals. Prove that $a/b+b/c+c/a=>(c+a)/(c+b) + (a+b)/(a+c) + (b+c)/(b+a)$

2014 Costa Rica - Final Round, 2
Tags: inequalities , algebra
Let $p_1,p_2, p_3$ be positive numbers such that $p_1 + p_2 + p_3 = 1$. If $a_1 <a_2 <a_3$ and $b_1 <b_2 <b_3$ prove that $$(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)$$

2001 South africa National Olympiad, 1
Tags: geometry , perimeter , inequalities , triangle inequality
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)

1999 Polish MO Finals, 1
Tags: inequalities , triangle inequality , geometry
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.

2005 Canada National Olympiad, 2
Tags: inequalities , number theory
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$. $a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$. $b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.

2016 India Regional Mathematical Olympiad, 3
Tags: number theory , inequalities , sum of digits
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.

2009 CentroAmerican, 6
Tags: modular arithmetic , inequalities , number theory
Find all prime numbers $ p$ and $ q$ such that $ p^3 \minus{} q^5 \equal{} (p \plus{} q)^2$.

1979 Chisinau City MO, 170
Tags: algebra , inequalities , recurrence relation
The numbers $a_1,a_2,...,a_n$ ( $n\ge 3$) satisfy the relations $$a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)$$ Prove that the numbers $a_1,a_2,...,a_n$ are non-negative.

1982 IMO Longlists, 35
Tags: geometry , circumcircle , inradius , trigonometry , geometric transformation , homothety , inequalities
If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

1992 Balkan MO, 3
Tags: inequalities , circumcircle , trigonometry , geometry
Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \] [i]Greece[/i]

2013 Dutch IMO TST, 5
Tags: algebra , inequalities
Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$

1996 Romania Team Selection Test, 13
Tags: inequalities
Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that \[ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. \] [i]Mircea Becheanu[/i]

2018 Romania Team Selection Tests, 3
Tags: floor function , algebra , inequalities
Given an integer $n \geq 2$ determine the integral part of the number $ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$

1994 Cono Sur Olympiad, 3
Tags: inequalities
Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$, knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$.

2010 Baltic Way, 19
Tags: inequalities , number theory
For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that \[p_1^2+p_2^2+\ldots +p_k^2=2010? \]