IMO 1985 LL FRA27

Origin: FRA

Problem

Let O be a point on the oriented Euclidean plane and (i, j) a directly oriented orthonormal basis. Let C be the circle of radius 1, centered at O. For every real number t and nonnegative integer n let Mn be the point on C for which ⟨i, −−−\to OMn⟩= cos 2nt (or −−−\to OMn = cos 2nti+sin 2ntj). Let k \geq2 be an integer. Find all real numbers t \in[0, 2\pi) that satisfy (i) M0 = Mk, and (ii) if one starts from M0 and goes once around C in the positive direction, one meets successively the points M0, M1, . . . , Mk−2, Mk−1, in this order.