Tetrahedron with Integer Sidelengths
Source: MEMO Team Competition, Quesiton 7
September 24, 2007
geometry3D geometrytetrahedrongeometric transformationreflectiongeometry proposed
Problem Statement
A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is and one of them is . Let be the sum of the sidelengths of the tetrahedron .
(a) Find all positive integers so that there exists a MEMO-Tetrahedron with l(T)\equal{}n.
(b) How many pairwise non-congruent MEMO-tetrahedrons satisfying l(T)\equal{}2007 exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).