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Brazil geometry

Problem

Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.
Solution
Use any coloring with all pairs at opposite ends of a diameter having opposite colors. For example, take to be a diameter. Color red, blue, all points in the interior of one arc red and all points in the interior of the other blue.

Techniques

Angle chasingColoring schemes, extremal arguments