2011 AIME II Problems/Problem 3
Contents
[hide]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
Solution 1
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)
Solution 2
You could also solve this problem with exterior angles. Exterior angles of any polygon add up to . Since there are
exterior angles in an 18-gon, the average measure of an exterior angles is
. We know from the problem that since the exterior angles must be in an arithmetic sequence, the median and average of them is
. Since there are even number of exterior angles, the middle two must be
and
, and the difference between terms must be
. Check to make sure the smallest exterior angle is greater than
:
. It is, so the greatest exterior angle is
and the smallest interior angle is
.