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4

Part of 1962 Swedish Mathematical Competition

Problems(1)

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

3/21/2021
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.
geometrynumber theorytrigonometryinequalities