# 2011 AIME II Problems/Problem 3

## Problem 3

The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

## Solution

The average angle in an 18-gon is . In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to . Thus for some positive (the sequence is increasing and thus non-constant) integer , the middle two terms are and . Since the step is the last term of the sequence is , which must be less than , since the polygon is convex. This gives , so the only suitable positive integer is 1. The first term is then $(160-17)^\circ = \fbox{143^\circ.}$ (Error compiling LaTeX. ! Missing $ inserted.)