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6 questions in one geometry problem, starting with a cube, old Vietnamese

Source: Vietnamese MO (VMO) 1971

August 23, 2018
geometry3D geometry

Problem Statement

ABCDA′B′C′D′ABCDA'B'C'D' is a cube (with ABCDABCD and A′B′C′D′A'B'C'D' faces, and AA′,BB′,CC′,DD′AA', BB', CC', DD' edges). LL is a line which intersects or is parallel to the lines AA′,BCAA', BC and DB′DB'. LL meets the line BCBC at MM (which may be the point at infinity). Let m=∣BM∣m = |BM|. The plane MAA′MAA' meets the line B′C′B'C' at EE. Show that ∣B′E∣=m|B'E| = m. The plane MDB′MDB' meets the line A′D′A'D' at FF. Show that ∣D′F∣=m|D'F| = m. Hence or otherwise show how to construct the point PP at the intersection of LL and the plane A′B′C′D′A'B'C'D'. Find the distance between PP and the line A′B′A'B' and the distance between PP and the line A′D′A'D' in terms of mm. Find a relation between these two distances that does not depend on mm. Find the locus of MM. Let SS be the envelope of the line LL as MM varies. Find the intersection of SS with the faces of the cube.
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