BelMath — Belarus Mathematics Olympiad

Problem: For positive integer $$m=2^k\cdot t,$$ where $k$ is a non-negative integer, $t$ is an odd integer, and let $$f(m)=t^{1-k}.$$ Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $$\prod_{m=1}^n f(m)$$ is a multiple of $a$.

— Dzianis Vistunou — IMO 2025