Question 115
Alice starts with some coins in a box. On the $n^{\text{th}}$ minute (for $n = 1, 2, 3, \dots$), she can choose to put an additional coin into the box or do nothing, and then she writes the number $2c^2 - n^2$ on the board where $c$ is the number of coins in the box. Show that for any positive integer $m$, there will eventually be two numbers on the board that sum to a multiple of $m$.