IMO 1977 LL FIN44

Origin: FIN

Problem

Let E be a finite set of points in space such that E is not contained in a plane and no three points of E are collinear. Show that E contains the vertices of a tetrahedron T = ABCD such that T \capE = {A, B, C, D} (including interior points of T ) and such that the projection of A onto the plane BCD is inside a triangle that is similar to the triangle BCD and whose sides have midpoints B, C, D.

Solution

Let d(X, \sigma) denote the distance from a point X to a plane \sigma. Let us consider the pair (A, \pi) where A \inE and \pi is a plane containing some three points B, C, D \inE such that d(A, \pi) is the smallest possi- ble. We may suppose that B, C, D are selected such that \triangleBCD con- tains no other points of E. Let A′ be the projection of A on \pi, and let lb, lc, ld be lines through B, C, D parallel to CD, DB, BC respec- tively. If A′ is in the half-plane determined by ld not containing BC, then d(D, ABC) \leqd(A′, ABC) < d(A, BCD), which is impossible. Sim- ilarly, A′ lies in the half-planes determined by lb, lc that contain D, and hence A′ is inside the triangle bordered by lb, lc, ld. The minimality prop- erty of (A, \pi) and the way in which BCD was selected guarantee that E \capT = {A, B, C, D}.