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Prove this inequality of radii of cicumsphere and insphere of a tetrahedron

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020
tetrahedron3D geometrycircumradiusinradiusinequalitiesgeometry

Problem Statement

In the any [ABCD][ABCD] tetrahedron we denote with α\alpha, β\beta, γ\gamma the measures, in radians, of the angles of the three pairs of opposite edges and with rr, RR the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality(3rR)3≤sin⁡α+β+γ3\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}(A refinement of inequality R≥3rR \geq 3r).
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