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 2 of 6: Encode later circles by angles

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In plain words

The angular offset is chosen to be exactly the extra radial distance, after multiplying by the current radius.

\theta_{i,j}=r_i(r_{i+j}-r_i),\quad 0<\theta_{i,j}<2\pi

Detailed analysis

Choose $x_{i,j}$ on $\mathcal C_i$ so that its angle from $OA$ is $\theta_{i,j}$ . The bound on the total sum gives $\theta_{i,j}<2\pi$ , and the defining radius formula then gives $C(x_{i,j})=\mathcal C_{i+j}$ .