## AIME I 2010 #10

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

### AIME I 2010 #10

Let be the number of ways to write in the form , where the 's are integers, and An example of such a representation is Find .
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stupidityismygam
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### Re: AIME I 2010 #10

stupidityismygam wrote:Let be the number of ways to write in the form , where the 's are integers, and An example of such a representation is Find .

Because, , and have a much greater range of possibilities. and are the coefficients that limit the total number .

For , has different possibilities, .

For , has different possibilities, .

For , has two different possilities, .

Hence, .

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### Re: AIME I 2010 #10

So, you're saying that always?
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AuSmith
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### Re: AIME I 2010 #10

AuSmith wrote:So, you're saying that always?

No. What he is saying is given an and an , there is only one possibility of what and can be.
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