Olympiad problems

1975 IMO, 1

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2004 Greece National Olympiad, 1

Tags: rearrangement inequality , inequalities

Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$ $x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$.

1999 IMC, 2

Tags: inequalities , rearrangement inequality

Does there exist a bijective map $f:\mathbb{N} \rightarrow \mathbb{N}$ so that $\sum^{\infty}_{n=1}\frac{f(n)}{n^2}$ is finite?

2010 Contests, 4

Tags: inequalities , rearrangement inequality

Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

2010 Contests, 1

Tags: inequalities , rearrangement inequality

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2012 Balkan MO, 2

Tags: trigonometry , function , triangle inequality , rearrangement inequality

Prove that \[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\] for all positive real numbers $x,y$ and $z$.

1978 IMO, 2

Tags: algebra , rearrangement inequality , inequality , summation

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

2011 ISI B.Stat Entrance Exam, 1

Tags: calculus , inequalities , rearrangement inequality , n-variable inequality

Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that \[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]