BALTIC-WAY 2024 Problem 8

Let $a, b, n$ be positive integers such that $a+b \leq n^{2}$. Alice and Bob play a game on an (initially uncoloured) $n \times n$ grid as follows:

• First, Alice paints $a$ cells green.
• Then, Bob paints $b$ other (i.e. uncoloured) cells blue.

Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of $a, b$ and $n$, who has a winning strategy.