Ali is hosting a large party. Together with his friends, people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of such that after some number of steps, the situation reaches a point where each person has exactly one apple.

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

Let be an arbitrary triangle and is the circumcenter of .Points lie on ,respectively such that the reflection of WRT is tangent to circumcircle of .Prove that the circumcircle of triangle is tangent to circumcircle of triangle .

Problem 1:

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.

Let be an odd prime number. Let be the sequence defined as follows: , and for , is the smallest integer greater than such that in there are no arithmetic progressions of length . We say that a positive integer is a ghost if it doesn’t appear in .
What is the smallest ghost that is not a multiple of ?

Proposed by Guerrero

For with , prove that

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