Nemotron-IMO-Bench 2026 A17

Let $n\geq 2$, and let $a_1,...,a_n$ be positive real numbers such that there are real numbers $c,d$ and $b_1,...,b_n$ with $$\sum_{i=1}^n \lfloor a_ix+b_i\rfloor=\lfloor cx+d\rfloor$$ for all $x\in \mathbb{R}$. Prove that $a_1,...,a_n$ are not pairwise distinct.