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Problem 904

AIME late

Geometry Difficulty 5.8 Prove it

Task 2. a) Prove that there exists a polygon that can be divided into two equal parts by a segment, where one end of the segment is the midpoint of a side of the polygon, and the other end divides some side in the ratio $1: 2$ .

b) Does there exist a convex polygon with this property?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution. We will show an example of a convex polygon with the desired property. The sides of the square $A B C D$ are divided in the ratio $1: 2$ by the points $M, N, P, Q$ :

The segments $M P$ and $N Q$ are bisected by the center of the square and divide the square into four equal quadrilaterals. The quadrilateral $A B N Q$ satisfies the condition of the problem.

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