1996 AIME Problems/Problem 3

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, we rewrite as $[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$ . 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 $(n+1)^2 \ge 1996$ , the smallest square after $1996$ is $2025 = 45^2$ , so our answer is $45 - 1 = \boxed{044}$ .

Alternatively, when $n = k$ , the exponents of $x$ or $y$ in $x^i y^i$ can be any integer between $0$ and $k$ inclusive. Thus, when $n=1$ , there are $(2)(2)$ terms and, when $n = k$ , there are $(k+1)^2$ terms. Therefore, we need to find the smallest perfect square that is greater than $1996$ . From trial and error, we get $44^2 = 1936$ and $45^2 = 2025$ . Thus, $k = 45\rightarrow n = \boxed{044}$ .

Solution FASTER

the floor of the square root of 1996 easily 44

1996 AIME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1996 AIME Problems/Problem 3

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