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PQ>1, sphere cirucmscribed on a regular tetrahedron

Source: Polish MO Recond Round 1986 p3

September 9, 2024
geometry3D geometryspheretetrahedrongeometric inequality

Problem Statement

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere S S is represented as the sum of four sets. Prove that one of these sets includes points P P , Q Q such that the length of the segment PQ PQ exceeds 1.
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