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
.
(b) How many pairwise non-congruent MEMO-tetrahedrons
satisfying
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).
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![l(T)](/media/m/f/d/0/fd0afa964801afa9e13b14962101f607.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
(a) Find all positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![l(T)=n](/media/m/3/5/8/358aa5a62c575781e127d502ca2deef4.png)
(b) How many pairwise non-congruent MEMO-tetrahedrons
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![l(T)=2007](/media/m/6/1/5/615f6705dc2854fc198bf1a38ece7435.png)