AUSTRIAN-POLISH 1999

Find the number of $6$ -tuples $(A_1,A_2,...,A_6)$ of subsets of $M = \{1,..., n\}$ (not necessarily different) such that each element of $M$ belongs to zero, three, or six of the subsets $A_1,...,A_6$ .

Find the best possible $k,k'$ such that $$k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'$$ for all positive reals $v,w,x,y,z$ .

Given an integer $n \ge 2$ , find all sustems of $n$ functions $f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 ... f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$ , $l$ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$ , $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$ . Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$ Prove that there exists at most one $n$ for which $a_n$ is a perfect square.

Solve in the nonnegative real numbers the system of equations $$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\, for \,\,\, n = 1,2,..., 1999 \ x_0 = x_{1999} \end{cases}$$

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$ .

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$ , (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.