4. Color all points on a circle either black or white. Ask: Is it necessarily true that there exists a triangle with vertices on the circle and all three vertices of the same color:
(1) Isosceles triangle; (2) Equilateral triangle;
(3) Rectangle; (4) Trapezoid?
Prove your conclusion.
(57th Belarusian Mathematical Olympiad)
Solution
(提示: If half of the circle is colored black and the other half is colored white, then
it is known that (2) and (3) do not hold. Considering a regular pentagon inscribed in the circle, by the pigeonhole principle, there must be three vertices of the same color, thus (1) holds. Considering each inscribed equilateral triangle, by the pigeonhole principle, there must be one side with both endpoints of the same color, thus, there exist infinitely many equal arcs with endpoints of the same color. Therefore, there must exist four points of the same color on the circle forming the vertices of a trapezoid.)
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.