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: 426

2010 Contests, 2
Tags: inequalities , inequalities unsolved
Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

2010 China Team Selection Test, 2
Tags: algebra , polynomial , inequalities , floor function , triangle inequality , inequalities unsolved
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

2004 China Girls Math Olympiad, 2
Tags: inequalities , inequalities unsolved
Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}. \]

2012 South East Mathematical Olympiad, 2
Tags: inequalities , inequalities unsolved
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.

2008 South africa National Olympiad, 3
Tags: inequalities , inequalities unsolved
Let $a,b,c$ be positive real numbers. Prove that \[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\] and determine when equality occurs.

2011 Uzbekistan National Olympiad, 1
Tags: inequalities , inequalities unsolved
Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$

2010 China Team Selection Test, 2
Tags: algebra , polynomial , inequalities , floor function , triangle inequality , inequalities unsolved
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

1976 IMO Longlists, 12
Tags: inequalities unsolved , inequalities
Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.

2005 China Team Selection Test, 2
Tags: inequalities , inequalities unsolved
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that \[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]

2006 Estonia National Olympiad, 1
Tags: inequalities unsolved , inequalities
Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real numbers x, for which this value is achieved.

2013 ELMO Shortlist, 9
Tags: inequalities , function , logarithms , inequalities unsolved
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that \[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]

2010 Contests, 4
Tags: inequalities unsolved , inequalities
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

2014 Saudi Arabia IMO TST, 1
Tags: inequalities , induction , inequalities unsolved
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?

1993 Hungary-Israel Binational, 3
Tags: inequalities , trigonometry , symmetry , function , vector , inequalities unsolved
Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

2012 Korea - Final Round, 1
Tags: inequalities unsolved , inequalities
Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

2010 BMO TST, 4
Tags: inequalities , LaTeX , inequalities unsolved
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.

2013 ELMO Problems, 2
Tags: inequalities , logarithms , function , calculus , derivative , inequalities unsolved , 3-variable inequality
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2011 Kosovo Team Selection Test, 1
Tags: inequalities , inequalities unsolved
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\]

1994 USAMO, 4
Tags: function , inequalities unsolved , inequalities
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]

2014 Saudi Arabia BMO TST, 3
Tags: inequalities , function , inequalities unsolved
Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that \[\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.\]

2005 China Team Selection Test, 3
Tags: inequalities , algebra , polynomial , trigonometry , induction , triangle inequality , inequalities unsolved
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

2005 MOP Homework, 2
Tags: inequalities , inequalities unsolved
Let $x$, $y$, $z$ be positive real numbers and $x+y+z=1$. Prove that $\sqrt{xy+z}+\sqrt{yz+x}+\sqrt{zx+y} \ge 1+\sqrt{xy}+\sqrt{yz}+\sqrt{zx}$.

1988 Polish MO Finals, 1
Tags: inequalities , function , algebra , domain , analytic geometry , inequalities unsolved
The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.

2013 International Zhautykov Olympiad, 3
Tags: inequalities , inequalities unsolved
Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that \[\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.\] [i]Proposed by Orif Ibrogimov, Uzbekistan.[/i]

1982 IMO Longlists, 28
Tags: inequalities , inequalities unsolved
Let $(u_1, \ldots, u_n)$ be an ordered $n$tuple. For each $k, 1 \leq k \leq n$, define $v_k=\sqrt[k]{u_1u_2 \cdots u_k}$. Prove that \[\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.\] ($e$ is the base of the natural logarithm).