3. Find conditions on the positive real number $a$ such that there exists a tetrahedron $k$ of whose edges $(k=1,2,3,4,5)$ have length $a$, and the other $6-k$ edges have length 1 .
3. We have several cases:
$1^{\circ} k=1$. W.l.o.g. let $A B=a$ and the remaining segments have length 1. Let $M$ be the midpoint of $C D$. Then $A M=B M=\sqrt{3} / 2(\triangle C D A$ and $\triangle C D B$ are equilateral) and $01$. Assume $A B=A C=A D=a$. Varying $A$ along the line perpendicular to the plane $B C D$ and through the center of $\triangle B C D$ we achieve all values of $a>1 / \sqrt{3}$. For $a\sqrt{2-\sqrt{3}}$.
$5^{\circ} k=5$. We reduce to $k=1$ and get $a>1 / \sqrt{3}$.