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(-1)^dP (-x), powerful + non-increasing polynomials with real coefficients

Source: SRMC 2020 P3 - Silk Road

August 18, 2020
algebrapolynomial

Problem Statement

A polynomial Q(x)=knxn+kn−1xn−1+…+k1x+k0 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 ∣k0∣=∣k1∣+∣k2∣+…+∣kn−1∣+∣kn∣ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | , and non-increasing , if k0≥k1≥…≥kn−1≥kn k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n . Let for the polynomial P(x)=adxd+ad−1xd−1+…+a1x+a0 P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 with nonzero real coefficients, where ad>0 a_d> 0 , the polynomial P(x)(x−1)t(x+1)s P (x) (x-1) ^ t (x + 1) ^ s is powerful for some non-negative integers s s and t t (s+t>0 s + t> 0 ). Prove that at least one of the polynomials P(x) P (x) and (−1)dP(−x) (- 1) ^ d P (-x) is nonincreasing.
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