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Problem 3 of Fourth round - 2 circles with radius r and 2r and a convex polygon

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

January 11, 2020
geometryrotationconvex polygon

Problem Statement

The angle of a rotation ρ\rho is α<180∘\alpha <180^\circ and ρ\rho maps the convex polygon MM in itself. Prove that there exist two circles c1c_1 and c2c_2 with radius rr and 2r2r, so that c1c_1 is inner for MM and MM is inner for c2c_2.
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