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 3 of 6: Build the point matrix

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In plain words

The matrix is a bookkeeping device: moving down one row and right one column always selects a circle farther out by the sum of the two indices.

M=(x_{i,j})_{i,j\ge1},\quad x_{i,j}\in\mathcal C_i,\quad C(x_{i,j})=\mathcal C_{i+j}

Detailed analysis

Arrange all these points in an infinite matrix, with row $i$ consisting of points on $\mathcal C_i$ . The entry in row $i$ , column $j$ points to the circle indexed by $i+j$ .