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forum.virtualchallengemeets.com • View topic - AMC 12A 2012 #7

AMC 12A 2012 #7

American Mathematics Contest & American Invitational Mathematics Examination. See for more information
Note:USAMO/IMO-level questions need to go in the Olympiad section.

AMC 12A 2012 #7

Postby nsguy1350 » Fri Jun 15, 2012 9:34 pm

Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
2013 District 1/Region:
NS - 319/355
MA - 340/332
CA - 294/287
SC - 344/292
CS - 212/124 (fail)
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Re: AMC 12A 2012 #7

Postby darksaber21 » Sat Jun 16, 2012 12:08 pm

If the integers are all in a sequence, and all add up to 360, then we can formulate:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Since we need a to be an integer, we start plugging in values in multiples of two. That's what I did anyway. Eventually, you'll hit negative numbers, so plug numbers for multiples of two for d until that occurs, so you can find the smallest number, which happens to be:
[unparseable or potentially dangerous latex formula]

Plug in a six for d, and you will get -3, so plug in a four to get your answer.
Now I know the stakes, and I'm willing to show what I gots. I will see you all at state for the first and final time, so let's end the year with a bang! :)
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