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Inradius of hexahedron and octahedron

Source: 1983 National High School Mathematics League, Exam One, Problem 11

February 21, 2020
geometry3D geometryoctahedroninradius

Problem Statement

For a regular hexahedron and a regular octahedron, all their faces are regular triangles, whose lengths of each side are aa. Their inradius are r1,r2r_1,r_2. r1r2=mn,gcd⁡(m,n)=1\frac{r_1}{r_2}=\frac{m}{n}, \gcd(m,n)=1. Then mn=mn=________.
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