# 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 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: | ||

− | + | 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. |

## Revision as of 21:21, 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 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.