Question 150
Polynomial $Q(x) = k_n x^n + k_{n-1} x^{n-1} + \dots + k_1 x + k_0$ with real coefficients is called *mighty* if $|k_0| = |k_1| + |k_2| + \dots + |k_{n-1}| + |k_n|$, and *non-increasing* if $k_0 \ge k_1 \ge \dots \ge k_{n-1} \ge k_n$.
Let $P(x) = a_d x^d + a_{d-1} x^{d-1} + \dots + a_1 x + a_0$ be a polynomial with real non-zero coefficients, such that $a_d \gt 0$ and $P(x)(x-1)^t (x+1)^s$ is *mighty* for some non-negative integers $s$ and $t$ ($s+t \gt 0$). Prove that at least one of the polynomials $P(x)$ and $(-1)^d P(-x)$ is *non-increasing*. (Navid Safaei, Iran)
Let $P(x) = a_d x^d + a_{d-1} x^{d-1} + \dots + a_1 x + a_0$ be a polynomial with real non-zero coefficients, such that $a_d \gt 0$ and $P(x)(x-1)^t (x+1)^s$ is *mighty* for some non-negative integers $s$ and $t$ ($s+t \gt 0$). Prove that at least one of the polynomials $P(x)$ and $(-1)^d P(-x)$ is *non-increasing*. (Navid Safaei, Iran)