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

2014 China Second Round Olympiad, 1
Tags: inequalities
Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]

2006 Vietnam Team Selection Test, 1
Tags: function , convex-concave inequalities , three variable inequality , inequalities
Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds: \[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \] When does the equality occur?

2015 China Team Selection Test, 2
Tags: inequalities , number theory
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 $.

2010 South East Mathematical Olympiad, 3
Tags: inequalities
Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$. Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$

2003 Putnam, 1
Tags: algorithm , partition , inequalities , floor function
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[n = a_1 + a_2 + \cdots a_k\] with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.

2009 Germany Team Selection Test, 3
Tags: algebra , inequalities
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

2001 District Olympiad, 1
Tags: algebra , inequalities
Consider the equation $x^2+(a+b+c)x+\lambda (ab+bc+ca)=0$ with $a,b,c>0$ and $\lambda\in \mathbb{R}$. Prove that: a)If $\lambda\le \frac{3}{4}$, the equation has real roots. b)If $a,b,c$ are the side lengths of a triangle and $\lambda\ge 1$, then the equation doesn't have real roots. [i]***[/i]

2017 China Team Selection Test, 5
Tags: inequalities
Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.

2009 Germany Team Selection Test, 3
Tags: inequalities , algebra
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

2012 Singapore Senior Math Olympiad, 5
Tags: inequalities , algebra
For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove \[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]

1997 Swedish Mathematical Competition, 1
Tags: geometric inequalities , inequalities , geometry
Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

2001 USA Team Selection Test, 7
Tags: rhombus , geometry , inequalities
Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and \[AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.\] Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

Estonia Open Senior - geometry, 2004.1.5
Tags: inequalities , area of triangle , geometric inequality , geometry
Find the smallest real number $x$ for which there exist two non-congruent triangles with integral side lengths having area $x$.

2023 NMTC Junior, P6
Tags: inequalities
The sum of squares of four reals $x,y,z,u$ is $1$. Find the minimum value of the expression $E=(x-y)(y-z)(z-u)(u-x)$. Find also the minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs.

2006 Macedonia National Olympiad, 3
Tags: inequalities
Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that \[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]

2020 German National Olympiad, 5
Tags: integer , inequalities
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$. Prove the inequality \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

1987 IMO Longlists, 32
Tags: number theory , inequalities
Solve the equation $28^x = 19^y +87^z$, where $x, y, z$ are integers.

2008 Bulgaria Team Selection Test, 2
Tags: trigonometry , complex number , inequalities , geometry
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

1978 Miklós Schweitzer, 8
Tags: inequalities , advanced fields
Let $ X_1, \ldots ,X_n$ be $ n$ points in the unit square ($ n>1$). Let $ r_i$ be the distance of $ X_i$ from the nearest point (other than $ X_i$). Prove that the inequality \[ r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4.\] [i]L. Fejes-Toth, E. Szemeredi[/i]

2010 Contests, 4
Tags: number theory , inequalities
Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2006 China Northern MO, 5
Tags: inequalities , function
$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]

2005 Slovenia Team Selection Test, 6
Tags: algebra , inequalities
Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$

1966 IMO Shortlist, 30
Tags: logarithm , inequalities , calculus
Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2009 Abels Math Contest (Norwegian MO) Final, 4b
Tags: algebra , sum , inequalities
Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

1984 IMO Longlists, 58
Tags: limit , algebra , inequalities
Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.