SILK-ROAD 2020 Problem 3

A polynomial $Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0$ with real coefficients is called powerful if the equality $| k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n |$ , and non-increasing , if $k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n$ . Let for the polynomial $P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0$ with nonzero real coefficients, where $a_d> 0$ , the polynomial $P (x) (x-1) ^ t (x + 1) ^ s$ is powerful for some non-negative integers $s$ and $t$ ( $s + t> 0$ ). Prove that at least one of the polynomials $P (x)$ and $(- 1) ^ d P (-x)$ is nonincreasing.