Nemotron-IMO-Bench 2026 C28

Let $k$ be an even positive integer, and let $p_1,...,p_k$ be pairwise distinct prime numbers. Finally, let $N=p_1...p_k$, $a,b\in \{1,2,...,N\}$ and let $S_1$ (resp. $S_2$) be the number of divisors $d$ of $N$ belonging to $[a,b]$ and having an even (resp. odd) number of prime factors. Prove that $S_1-S_2\leq \binom{k}{k/2}$.