For each of
![k=1,2,3,4,5](/media/m/c/3/1/c311e9fc6c87e8b2255b78f9cf353e8a.png)
find necessary and sufficient conditions on
![a>0](/media/m/4/1/b/41bf6a8eeba84545ed84e7cbaea7fbcc.png)
such that there exists a tetrahedron with
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
edges length
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and the remainder length
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
.
%V0
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.