Difference between revisions of "2011 AIME II Problems/Problem 3"

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Problem:
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== 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.
 
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 ==
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>
 
 
Set up an equation where ''x'' is the measure of the smallest angle, and ''y'' is the increase in angle measure.
 
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
 
Solving the equation for integer values (or a formula that I don't know) you get ''x''=7, and ''y''=18
 
The smallest angle is therefore 7.
 
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.
 
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. ! Missing $ inserted.)

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