Track / Stage 4 / 11 of 340 #271 of 1964

Problem 271

AMC 12 late, AIME early

Geometry Difficulty 4.2 Prove it Brazilian Mathematical Olympiad · 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 $x$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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 $2L$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.