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min value of (cos ax + cos bx) <= r < 0

Source: 1968 Swedish Mathematical Competition p5

March 21, 2021
trigonometryalgebramininequalities

Problem Statement

Let a,ba, b be non-zero integers. Let m(a,b)m(a, b) be the smallest value of cos⁡ax+cos⁡bx\cos ax + \cos bx (for real xx). Show that for some rr, m(a,b)≤r<0m(a, b) \le r < 0 for all a,ba, b.
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