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Problem 1132

AIME late

Combinatorics Difficulty 5.1 Prove it All-Soviet-Union Mathematical Olympiad · Soviet Union

An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$ . A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.

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

Solution:

The diagram has $1 + 2 + \ldots + n = n(n + 1) / 2$ upright triangles and $1 + 2 + \ldots + n - 1 = n(n - 1) / 2$ upside down triangles. It has $1 + 2 + \ldots + n + 1 = (n + 1)(n + 2) / 2$ vertices. So the path must be $(n + 1)(n + 2) / 2 - 1 = (n^2 + 3n) / 2$ units long. Each unit length of the path is in just one upright triangle. The path cannot contain all three sides of a small triangle, or it would pass through a vertex more than once. So it must contain two sides of $(n^2 + 3n) / 2 - n(n + 1) / 2 = n$ triangles. But if it contains two sides of a triangle, then it must make an acute angle at the vertex where they meet.

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