IL-GROSSMAN 1995

Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$ . Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$ . Then player $II$ chooses another square and marks it with $X$ . They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.

Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient?

Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$ .