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2019 Jozsef Wildt International Math Competition, W. 58
Tags: insphere , 3d geometry , exsphere , circumsphere , inequalities , tetrahedron
In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$