PL-OM 1990

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that

$$\sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1$$

Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$ .

In a tournament, every two of the $n$ players played exactly one match with each other (no draws). Prove that it is possible either (i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$ ; or (ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.

A triangle whose all sides have length not smaller than $1$ is inscribed in a square of side length $1$ . Prove that the center of the square lies inside the triangle or on its boundary.

Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$ Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$ .

Prove that for all integers $n > 2$ , $$3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i}$$