### MEMO 2007 ekipno problem 7

Kvaliteta:

Avg: 2,3Težina:

Avg: 6,0 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).

(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).

Izvor: Srednjoeuropska matematička olimpijada 2007, ekipno natjecanje, problem 7