10. [15] 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.
Answer: 26
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.