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f(x) =x^2 + 2bx + c >= for all integers n, then f(x) >= 0

Source: Germany Federal - Bundeswettbewerb Mathematik 1990, round 1, p1

February 21, 2020
trinomialquadratic polynomialInteger Polynomialalgebra

Problem Statement

Consider the trinomial f(x)=x2+2bx+cf(x) = x^2 + 2bx + c with integer coefficients bb and cc. Prove that if f(n)≥0f(n) \ge 0 for all integers nn, then f(x)≥0f(x) \ge 0 even for all rational numbers xx.
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