Difference between revisions of "2011 AIME II Problems/Problem 3"
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− | == Problem | + | ==Problem== |
The degree measures of the angles in a [[convex polygon|convex]] 18-sided polygon form an increasing [[arithmetic sequence]] with integer values. Find the degree measure of the smallest [[angle]]. | The degree measures of the angles in a [[convex polygon|convex]] 18-sided polygon form an increasing [[arithmetic sequence]] with integer values. Find the degree measure of the smallest [[angle]]. | ||
− | + | ==Solution== | |
− | == Solution == | ||
===Solution 1=== | ===Solution 1=== | ||
The average angle in an 18-gon is <math>160^\circ</math>. In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to <math>160^\circ</math>. Thus for some positive (the sequence is increasing and thus non-constant) integer <math>d</math>, the middle two terms are <math>(160-d)^\circ</math> and <math>(160+d)^\circ</math>. Since the step is <math>2d</math> the last term of the sequence is <math>(160 + 17d)^\circ</math>, which must be less than <math>180^\circ</math>, since the polygon is convex. This gives <math>17d < 20</math>, so the only suitable positive integer <math>d</math> is 1. The first term is then <math>(160-17)^\circ = \fbox{143}.</math> | The average angle in an 18-gon is <math>160^\circ</math>. In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to <math>160^\circ</math>. Thus for some positive (the sequence is increasing and thus non-constant) integer <math>d</math>, the middle two terms are <math>(160-d)^\circ</math> and <math>(160+d)^\circ</math>. Since the step is <math>2d</math> the last term of the sequence is <math>(160 + 17d)^\circ</math>, which must be less than <math>180^\circ</math>, since the polygon is convex. This gives <math>17d < 20</math>, so the only suitable positive integer <math>d</math> is 1. The first term is then <math>(160-17)^\circ = \fbox{143}.</math> | ||
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==See also== | ==See also== | ||
{{AIME box|year=2011|n=II|num-b=2|num-a=4}} | {{AIME box|year=2011|n=II|num-b=2|num-a=4}} | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 00:47, 10 March 2020
Problem
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
Solution 2
Another way to solve this problem would be to use 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
.
See also
2011 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.