Olympiad problems

2007 China Team Selection Test, 1

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

2014 Iran MO (3rd Round), 4

Tags: inequalities , inequalities proposed

For any $a,b,c>0$ satisfying $a+b+c+ab+ac+bc= 3$, prove that \[2\leq a+b+c+abc\leq 3\] [i]Proposed by Mohammad Ahmadi[/i]

2006 Balkan MO, 1

Tags: inequalities , rearrangement inequality , inequalities theorems , inequalities proposed

Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]

2023 Indonesia TST, A

Tags: algebra , inequalities proposed , Indonesia , Indonesia TST

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

2008 Pan African, 1

Tags: inequalities , rearrangement inequality , inequalities proposed

Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.

Oliforum Contest IV 2013, 5

Tags: number theory , greatest common divisor , inequalities , quadratics , inequalities proposed

Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)

2006 Taiwan National Olympiad, 1

Tags: inequalities , inequalities proposed

Positive reals $a,b,c$ satisfy $abc=1$. Prove that $\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.

2008 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , inequalities proposed

Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that: \[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}. \]

2012 Iran Team Selection Test, 1

Tags: inequalities , inequalities proposed

For positive reals $a,b$ and $c$ with $ab+bc+ca=1$, show that \[\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}\] [i]Proposed by Morteza Saghafian[/i]

1992 Hungary-Israel Binational, 1

Tags: inequalities , induction , function , inequalities proposed

Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then \[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]

1983 Vietnam National Olympiad, 2

Tags: inequalities proposed , inequalities

Decide whether $S_n$ or $T_n$ is larger, where \[S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}\]

2024 HMIC, 2

Tags: Inequality , algebra , inequalities proposed

Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/i]

2008 Tournament Of Towns, 5

Tags: inequalities , inequalities proposed

Let $a_1,a_2,\cdots,a_n$ be a sequence of positive numbers, so that $a_1 + a_2 +\cdots + a_n \leq \frac 12$. Prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.\] [hide="Remark"]Remark. I think this problem was posted before, but I can't find the link now.[/hide]

2013 Serbia National Math Olympiad, 6

Tags: inequalities , inequalities proposed

Find the largest constant $K\in \mathbb{R}$ with the following property: if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]

2013 Czech-Polish-Slovak Match, 2

Tags: inequalities , inequalities proposed

Prove that for every real number $x>0$ and each integer $n>0$ we have \[x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)\]

2008 Saint Petersburg Mathematical Olympiad, 4

Tags: induction , inequalities proposed , inequalities

The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.

1994 Baltic Way, 3

Tags: trigonometry , inequalities proposed , inequalities

Find the largest value of the expression \[xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}\]

2009 IberoAmerican Olympiad For University Students, 3

Tags: inequalities proposed , inequalities

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$. Find the maximum value of $f$.

2011 Iran MO (3rd Round), 2

Tags: inequalities , inequalities proposed

For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$. Find the minimum of $x^2+y^2+z^2+t^2$. [i]proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi[/i]

2009 Indonesia TST, 2

Tags: inequalities , inequalities proposed

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

2011 Bosnia Herzegovina Team Selection Test, 2

Tags: inequalities , inequalities proposed

Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality \[a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1\] holds.

2005 Korea - Final Round, 2

Tags: inequalities , induction , quadratics , AMC , USA(J)MO , USAMO , inequalities proposed

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , \[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]

2010 Mediterranean Mathematics Olympiad, 2

Tags: inequalities , inequalities proposed , Mediterranean , n-variable inequality

Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality \[ \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\] does holds.

2008 China Team Selection Test, 2

Tags: inequalities proposed , inequalities

For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$

2024 Czech-Polish-Slovak Junior Match, 4

Tags: inequalities , inequalities proposed , Integer

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.