Problem

Source: Stars of Mathematics 2024 P2 (senior level)

Tags: number theory, double factorial



For any positive integer $n$ we define $n!!=\prod_{k=0}^{\lceil n/2\rceil -1}(n-2k)$. Prove that if the positive integers $a,b,c$ satisfy $a!=b!!+c!!$, then $b$ and $c$ are odd. Proposed by Mihai Cipu