Discrete Mathematics
# Pigeonhole Principle

Find the largest integer $n$ that satisfies the following condition:

If any 6 points are chosen on the perimeter of a circle, then we can draw semicircle of the circle, such that there are at least $n$ points on it.

Note: A point on the perimeter of the semicircle is considered to lie on it.

Alice and Betty play a game where Alice goes first. They each say a distinct integer from 1 to 15 (inclusive). The first person to say an integer which, when summed with a previously spoken integer, gives the value of 16, will lose the game.

Who will **win** the game?