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Difference between revisions of "2011 AIME II Problems/Problem 3"
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Problem:
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== Problem 3 ==
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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.
 
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.
  
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== Solution ==
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Solution:
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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^\circ.}</math>
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Set up an equation where ''x'' is the measure of the smallest angle, and ''y'' is the increase in angle measure.
 
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You get 18''x''+153''y''=2880, because (x+0y)+(x+y)+(x+2y)+...(x+17y)=18''x''+153''y''=the total angle measures of all of the angles in an 18-gon=2880
 
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Solving the equation for integer values (or a formula that I don't know) you get ''x''=7, and ''y''=18
 
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The smallest angle is therefore 7.
 
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However, we aren't done here. The smallest possible angle for a 18-gon with an arithmetic sequence is 7 degrees, we also need ''x''+17''y''<180 because it is convex.  By working down from (7,18) to (24,16) etc. we get to the final possibility (143, 2) which satisfies ALL of the requirements.
 
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The smallest angle is therefore 143
 

Revision as of 00:30, 2 April 2011

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)

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