MathDB

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.

Problem Statement