Track / Stage 5 / 228 of 400 #828 of 1964

Problem 828

AIME late

Geometry Difficulty 5.6 Prove it

8.3. Can five points $A, B, C, D, E$ be marked (and labeled) on a plane so that the five triangles $A B C, B C D, C D E, D E A, E A B$ are acute-angled?

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

Answer: It is possible. Solution. Consider a regular pentagon, mark its vertices and denote them as shown in the figure (moving clockwise and marking every other vertex). Then all five triangles will be isosceles and equal to each other (which means they are all acute-angled; it is not difficult to calculate their angles: $\left.36^{\circ}, 72^{\circ}, 72^{\circ}\right)$ ).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.