ILL 1985 Problem 27

Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$ , centered at $O$ . For every real number $t$ and non-negative integer $n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$ ). Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy**(i)** $M_0 = M_k$ , and**(ii)** if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$ , in this order.