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IMO LongList 1967, Romania 4

Source: IMO LongList 1967, Romania 4

December 16, 2004
trigonometryalgebraTrigonometric EquationsgeometryIMO ShortlistIMO Longlist

Problem Statement

(i) Solve the equation: sin⁡3(x)+sin⁡3(2π3+x)+sin⁡3(4π3+x)+34cos⁡2x=0. \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0. (ii) Supposing the solutions are in the form of arcs ABAB with one end at the point AA, the beginning of the arcs of the trigonometric circle, and PP a regular polygon inscribed in the circle with one vertex in AA, find: 1) The subsets of arcs having the other end in BB in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end BB in one of the vertices of polygon PP whose number of sides is prime or having factors other than 2 or 3.
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