ILL 1966 Problem 23
Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle.
(a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right.
(b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.
Dataset text AoPS wiki problems. Quoted with source.
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