ÖMO 2016

Consider a board consisting of $n\times n$ unit squares where $n \ge 2$ . Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists. (a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game. (b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game.

Proposed by Theresia Eisenkölbl

Let $a,b,c$ be three integers for which the sum $$\frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$$ is integer. Prove that each of the three numbers $$\frac{ab}{c}, \frac{ac}{b}, \frac{bc}{a}$$ is integer.

(Proposed by Gerhard J. Woeginger)