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.  Second Day (July 11)
3. We have several cases: $1^{\circ} k=1$. W.l.o.g. let $AB=a$ and the remaining segments have length 1. Let $M$ be the midpoint of $CD$. Then $AM=BM=\sqrt{3} / 2$ (triangles $CDA$ and $CDB$ are equilateral) and $01$. Assume $AB=AC=AD=a$. Varying $A$ along the line perpendicular to the plane $BCD$ and through the center of $\triangle BCD$ 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 $a>1 / \sqrt{3}$.