ÖMO 1999

Prove that for each positive integer $n$ , the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$ . The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$ , respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$ . Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$ . Describe the locus of the intersection point.

Find all pairs $(x, y)$ of real numbers such that $$y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999$$ where $f(x)=[x]$ is the floor function.

Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$ .

Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$ , find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$ .

Two players $A$ and $B$ play the following game. An even number of cells are placed on a circle. $A$ begins and $A$ and $B$ play alternately, where each move consists of choosing a free cell and writing either $O$ or $M$ in it. The player after whose move the word $OMO$ (OMO = Osterreichische Mathematik Olympiade) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player $B$ plays correctly, then player $A$ cannot win.