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10 from 64 points in unit circle, lies on circle of radius 1/2

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1982 p4

September 10, 2024
geometrycombinatoricscombinatorial geometrypoints

Problem Statement

In a circle with a radius of 11, 6464 mutually different points are selected. Prove that 1010 mutually different points can be selected from them, which lie in a circle with a radius 12\frac12.