Olympiad problems

2002 Greece National Olympiad, 1

The real numbers $a,b,c$ with $bc\neq0$ satisfy $\frac{1-c^2}{bc}\geq0.$ Prove that $10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.$

2003 Federal Competition For Advanced Students, Part 1, 2

Tags: inequalities , geometry , inequalities proposed

Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

2009 China Second Round Olympiad, 2

Tags: logarithms , inequalities proposed , inequalities

Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

2011 All-Russian Olympiad Regional Round, 11.8

Tags: inequalities , inequalities proposed

$b$ and $c$ are positive. Prove the inequality \[ \left(b-c\right)^{2011}\left(b+c\right)^{2011}\left(c-b\right)^{2011} \geq \left(b^{2011}-c^{2011}\right)\left(b^{2011}+c^{2011}\right)\left(c^{2011}-b^{2011}\right). \] (Author: V. Senderov)

2011 Switzerland - Final Round, 6

Tags: inequalities , inequalities proposed

Let $a, b, c, d$ be positive real numbers satisfying $a+b+c+d =1$. Show that \[\frac{2}{(a+b)(c+d)} \leq \frac{1}{\sqrt{ab}}+ \frac{1}{\sqrt{cd}}\mbox{.}\] [i](Swiss Mathematical Olympiad 2011, Final round, problem 6)[/i]

1982 Dutch Mathematical Olympiad, 1

Tags: inequalities , inequalities proposed

Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?

2011 Singapore Senior Math Olympiad, 5

Tags: inequalities , inequalities proposed

Given $x_1,x_2,\dots,x_n>0,n\geq 5$, show that \[\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}\]

2025 Czech-Polish-Slovak Junior Match., 5

Tags: algebra , inequalities proposed

For every integer $n\geq 1$ prove that $$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$

2016 Bulgaria National Olympiad, Problem 3

Tags: inequalities , algebra , inequalities proposed

For $a,b,c,d>0$ prove that $$\frac {a+\sqrt{ab}+\sqrt[3]{abc}+\sqrt[4]{abcd}}{4} \leq \sqrt[4]{a.\frac{a+b}{2}.\frac{a+b+c}{3}.\frac{a+b+c+d}{4}}$$

2002 India IMO Training Camp, 20

Tags: inequalities , inequalities proposed

Let $a,b,c$ be positive real numbers. Prove that \[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]

2011 ELMO Shortlist, 5

Tags: inequalities , geometry , inequalities proposed

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]

2001 Irish Math Olympiad, 4

Tags: inequalities , inequalities proposed

Prove that for all positive integers $ n$: $ \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}$.

1989 IberoAmerican, 2

Tags: inequalities , trigonometry , inequalities proposed

Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality \[\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\]

1997 Estonia National Olympiad, 2

Tags: inequalities , inequalities proposed

Let $x$ and $y$ be real numbers. Show that\[x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}.\]

2011 Romania Team Selection Test, 2

Tags: inequalities , inequalities proposed , Balkan

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

1979 IMO Longlists, 35

Tags: geometry , induction , area of a triangle , Heron's formula , inequalities proposed , inequalities

Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

2011 ELMO Shortlist, 4

Tags: inequalities , percent , induction , inequalities proposed

In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

2002 Poland - Second Round, 3

Tags: inequalities , inequalities proposed

Find all positive integers $n$ such that for all real numbers $x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n$ the following inequality holds: \[ x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot \]

2013 Uzbekistan National Olympiad, 1

Tags: inequalities , geometry , geometric transformation , reflection , inequalities proposed

Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]

2013 India IMO Training Camp, 1

Tags: inequalities , induction , rearrangement inequality , inequalities proposed

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that \[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]

1992 Romania Team Selection Test, 9

Tags: inequalities , inequalities proposed

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$. [i]Laurențiu Panaitopol[/i] and [i]Ioan Tomescu[/i]

1998 Turkey MO (2nd round), 2

Tags: inequalities , search , inequalities proposed

If $0\le a\le b\le c$ real numbers, prove that $(a+3b)(b+4c)(c+2a)\ge 60abc$.

2013 China Team Selection Test, 2

Tags: inequalities proposed , inequalities , China TST

Let $k\ge 2$ be an integer and let $a_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n$ be non-negative real numbers. Prove that\[\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i\equal{}1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i\equal{}1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.\]

2004 Mediterranean Mathematics Olympiad, 3

Tags: trigonometry , inequalities , inequalities proposed

Let $a,b,c>0$ and $ab+bc+ca+2abc=1$ then prove that \[2(a+b+c)+1\geq 32abc\]

2001 Italy TST, 2

Tags: inequalities , inequalities proposed , algebra

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]