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 $160^\circ$ . In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to $160^\circ$ . Thus for some positive (the sequence is increasing and thus non-constant) integer $d$ , the middle two terms are $(160-d)^\circ$ and $(160+d)^\circ$ . Since the step is $2d$ the last term of the sequence is $(160 + 17d)^\circ$ , which must be less than $180^\circ$ , since the polygon is convex. This gives $17d < 20$ , so the only suitable positive integer $d$ is 1. The first term is then $(160-17)^\circ = \fbox{143^\circ.}$ (Error compiling LaTeX. Unknown error_msg)

Page

Toolbox

Search

2011 AIME II Problems/Problem 3

Problem 3

Solution