1996 AIME Problems/Problem 3

Revision as of 20:54, 24 September 2007 by 1=2 (talk | contribs) (Solution)

Problem

Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms.

Solution

Using Simon's Favorite Factoring Trick, rewrite as $[(x-3)(y-7)]^n = (x-3)^n(y-7)^n$. Both binomial expansions will contain $n+1$ non-like terms; their product will contain $(n+1)^2$ terms, as each term will have an unique power of $x$ or $y$ and so none of the terms will need to be collected. Hence $(n+1)^2 > 1996$, the smallest square after $1996$ is $2025 = 45^2$, so our answer is $45 - 1 = 044$.

See also

1996 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions