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

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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
 
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.
 
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.
+
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
 
The smallest angle is therefore 143

Revision as of 07:14, 31 March 2011

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:

Set up an equation where x is the measure of the smallest angle, and y is the increase in angle measure. You get 18x+153y=2880, because (x+0y)+(x+y)+(x+2y)+...(x+17y)=18x+153y=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+17y<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

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