Problem 3

For each

k=1,2,3,4,5

, find necessary and sufficient conditions on

a>0

for the existence of a tetrahedron with

k

edges of length

a

and the remaining

6-k

edges of length

1

Step 1 of 5: Handle one and five long edges

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In plain words

Hinging two rigid triangles changes one distance continuously; the flat positions are exactly the excluded degenerate endpoints.

0<a<\sqrt3\quad(k=1),\qquad a>\frac1{\sqrt3}\quad(k=5)

A tetrahedron as an illustrative four-vertex, six-edge model; the contest edge lengths are not to scale.

Detailed analysis

For $k=1$ , let the unique $a$ -edge be $AB$ and hinge the unit equilateral triangles $ACD$ and $BCD$ about $CD$ . The distance $AB$ varies strictly between $0$ and $\sqrt3$ in nondegenerate positions, giving $0<a<\sqrt3$ . Swapping the roles of $a$ and $1$ and scaling by $1/a$ gives $k=5$ : $1/a<\sqrt3$ , or $a>1/\sqrt3$ .