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geometri inequaliy with circles and a regular octahedron

Source: 1984 Polish MO Finals p3

February 25, 2020

geometryinequalitiesoctahedron3D geometrycircles

Problem Statement

Let

W

be a regular octahedron and

O

be its center. In a plane

P

containing

O

circles

k_1(O,r_1)

and

k_2(O,r_2)

are chosen so that

k_1 \subset P\cap W \subset k_2

. Prove that

\frac{r_1}{r_2}\le \frac{\sqrt3}{2}