Let be an integer. In a configuration of an board, each of the cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of , the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey

There are positive integers such that for every index , with , is a multiple of . Determine the greatest possible value of the largest of the numbers.

Let and be positive integers satisfying the equation . If is a prime number, prove that the number is composite.

Proposed by Serbia

A rectangle with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.

Proposed by Jeck Lim, Singapore

Let be the set of real numbers. Determine all functions such that for all pairs of real numbers and .