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f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0

Source: 1964 Swedish Mathematical Competition p5

March 21, 2021
trigonometryminmaxinequalitiesalgebra

Problem Statement

a1,a2,...,ana_1, a_2, ... , a_n are constants such that f(x)=1+a1cosx+a2cos2x+...+ancosnx≥0f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0 for all xx. We seek estimates of a1a_1. If n=2n = 2, find the smallest and largest possible values of a1a_1. Find corresponding estimates for other values of nn.
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