IMO 1986 LL CZS17

Origin: CZS

Problem

We call a tetrahedron right-faced if each of its faces is a right- angled triangle. (a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra. (b) Prove that a tetrahedron with vertices A1, A2, A3, A4 is fight-faced if and only if there exist four distinct real numbers c1, c2, c3, and c4 such that the edges AjAk have lengths AjAk = |cj −ck| for 1 \leqj < k \leq4.