Question 5
Alice starts with some coins in a box. On the $n$th minute (for $n=1,2,3,\ldots$), 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 $m$.