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 6 of 6: Exhaust all colors

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In plain words

Finite color supply cannot support an endless sequence of newly forbidden colors; the contradiction forces the desired match.

c_1,\ldots,c_n\text{ all forbidden on one infinite family of circles}\ \Rightarrow\ \text{contradiction}

Detailed analysis

After the number $n$ of available colors, the induction gives a circle in the surviving infinite family on which none of the $n$ colors can occur. This is impossible because every point of the plane has one of those colors. Therefore the initial assumption was false, and the required off-line point exists.