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inequality for polynomial in Z[x]

Source: 2001 Moldova MO Grade 11 P4

April 13, 2021
inequalitiesalgebrapolynomial

Problem Statement

Let P(x)=xn+a1xn−1+…+anP(x)=x^n+a_1x^{n-1}+\ldots+a_n (n≥2n\ge2) be a polynomial with integer coefficients having nn real roots b1,…,bnb_1,\ldots,b_n. Prove that for x0≥max⁡{b1,…,bn}x_0\ge\max\{b_1,\ldots,b_n\}, P(x0+1)(1x0−b1+…+1x0−bn)≥2n2.P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.
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