HMMT February

Geometry Difficulty 4.9 AIME Find the answer

Suppose there exists a convex $n$ -gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We can't have $n=3$ since the sum of the angles must be $180^{\circ}$ but the sum of three odd numbers is odd. On the other hand, for $n=4$ we can take a quadrilateral with angle measures $83^{\circ}, 83^{\circ}, 97^{\circ}, 97^{\circ}$ . The largest possible value of $n$ is 360. For larger $n$ we can't even have all angles have integer measure, and 179 happens to be prime. So, the answer is $360-4=356$ .

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