Problem

Source: BxMO 2024 P2

Tags: combinatorics



Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$