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. Unknown error_msg)