Competition problem — Geometry — Maths Olympiad Prep

Geometry Difficulty 4.2 AIME Prove it Brazil

A regular tetrahedron has side $L$ . What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$ ?

Solution

The answer is $2L$ . Consider the following net of the tetrahedron:

Let $P$ be a point of one of the edges of the tetrahedron. The loop will pass through $P$ some time. But $P'$ on the net coincides with $P$ on the tetrahedron, so by the triangular inequality the loop must be at least $PP' = 2L$ long.

It is possible to maintain the loop on the net parallel to $AA'$ , so the least value is $2L$ .

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