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Cubes, surjections, and isometries

Source: Poland Math Olympiad 1992 Round 1 #10

June 2, 2023
geometry3D geometry

Problem Statement

Let CC be a cube and let f:C⟶Cf: C \longrightarrow C be a surjection with ∣PQ∣≥∣f(P)f(Q)∣|PQ| \geq |f(P)f(Q)| for all P,Q∈CP,Q \in C. Prove that ff is an isometry.
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