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f(2) >= 3^n for polynomial with n real roots and non-negative coefficients

Source: 1983 Hungary - Kürschák Competition p2

October 10, 2022
algebrapolynomial

Problem Statement

Prove that f(2)≥3nf(2) \ge 3^n where the polynomial f(x)=xn+a1xn−1+...+an−1x+1f(x) = x_n + a_1x_{n-1} + ...+ a_{n-1}x + 1 has non-negative coefficients and nn real roots.
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