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2 lines intersect in pairs, n=a^4+b^4, a_1 cosx + a_2 cos2x +... +a_n cosnx > 0

Source: 1962 Swedish Mathematical Competition p4

March 21, 2021
geometrynumber theorytrigonometryinequalities

Problem Statement

Which of the following statements are true? (A) XX implies YY, or YY implies XX, where XX is the statement, the lines L1,L2,L3L_1, L_2, L_3 lie in a plane, and YY is the statement, each pair of the lines L1,L2,L3L_1, L_2, L_3 intersect. (B) Every sufficiently large integer nn satisfies n=a4+b4n = a^4 + b^4 for some integers a, b. (C) There are real numbers a1,a2,...,ana_1, a_2,... , a_n such that a1cos⁡x+a2cos⁡2x+...+ancos⁡nx>0a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0 for all real xx.
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