Difference between revisions of "2011 AIME II Problems/Problem 2"
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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 | 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 | 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. |
Revision as of 22:13, 30 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.
