KöMaL problem A, 2026

Let $a_0=0<a_1<a_2<\ldots<a_n$ be integers such that the sequence $b_k=\frac{a_{k+1}-a_k}{2k+1}$ ($k=0$, 1, $\ldots$, $n-1$) is non-decreasing. Suppose that $c_1$, $c_2$, $\ldots$, $c_n$ are real numbers such that the polynomial $1+\sum_{k=1}^n c_kx^{a_k}$ is divisible by the polynomial $(x+1)^n$. Show that $2>|c_1|>|c_2|>\ldots>|c_n|$.