Competition problem — Geometry — Maths Olympiad Prep

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Suppose we have an $n$ -gon such that each interior angle, measured in degrees, is a positive integer. Suppose further that all angles are less than $180^{\circ}$ , and that all angles are different sizes. What is the maximum possible value of $n$ ? Prove your answer.

Solution

Solution:

Let's work with the exterior angles (each is $180$ minus the interior angle). Then the conditions on the exterior angles are identical to the conditions on the interior angles: each is a positive integer between $1$ and $179$ inclusive. The sum of the exterior angles is exactly $360$ .

However, the sum of $1$ through $27$ is $27 \cdot 28 / 2 = 378$ , which is too large. We can get $26$ using angles of $1$ through $25$ (sum $325$ ) and an angle of $35$ . The previous problem shows that this is actually possible.

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