Problem 3

Given points

O

and

A

in the plane, every point is colored with one of a finite number of colors. For a point

X

, let

C(X)

be the circle centered at

O

with radius

OX+\frac{\angle AOX}{OX}

, where

\angle AOX

is measured in radians in

\left[0,2\pi\right)

. Prove that there is a point

X

, not on

OA

, such that the color of

X

appears on the circumference of

C(X)

Step 1 of 6: Choose nested radii

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In plain words

The radii grow forever but remain in a bounded range, leaving enough angular room for every later circle to be encoded.

a_n=2^{-n-1},\quad r_i=\sum_{t=1}^{i}a_t,\quad \left(\sum_{n\ge1}a_n\right)^2<2\pi

The reference ray OA and a sample polar angle.

Detailed analysis

Take any positive sequence with squared total sum less than $2\pi$ , for example the displayed geometric sequence. Let the circle of radius $r_i$ centered at $O$ be $\mathcal C_i$ .