IMO 1967 LL ROM45

Origin: ROM

Problem

(a) Solve the equation sin3 x + sin3 2\pi 3 + x 

• sin3 4\pi 3 + x 
• 3 4 cos 2x = 0. (b) Suppose the solutions are in the form of arcs AB of the trigonometric circle (where A is the beginning of arcs of the trigonometric circle), and P is a regular n-gon inscribed in the circle with one vertex at A. (1) Find the subset of arcs with the endpoint B at a vertex of the regular dodecagon. (2) Prove that the endpoint B cannot be at a vertex of P if 2, 3 ∤n or n is prime.

Solution

(a) Using the formula 4 sin3 x = 3 sin x −sin 3x one can easily reduce the given equation to sin 3x = cos 2x. Its solutions are given by x = (4k + 1)\pi/10, k \inZ. (b) (1) The point B corresponding to the solution x = (4k + 1)\pi/10 is a vertex of the regular dodecagon if and only if (4k + 1)\pi/10 = 2m\pi/12, i.e., 3(4k + 1) = 5m for some m \inZ. This is possible if and only if 5 | 4k + 1, i.e., k \equiv1 (mod 5). (2) Similarly, if the point B corresponding to x = (4k + 1)\pi/10 is a vertex of a polygon P, then (4k + 1)n = 20m for some m \inN, which implies that 4 | n.