A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$. (a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)=n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)=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).
Problem
Source: MEMO Team Competition, Quesiton 7
Tags: geometry, 3D geometry, tetrahedron, geometric transformation, reflection, geometry proposed