Difference between revisions of "1996 AIME Problems/Problem 3"

Revision as of 18:31, 4 July 2013

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 $45 - 1 = \boxed{044}$ .

Revision as of 18:27, 2 June 2012 (view source) Raninf (talk \| contribs) m (Just some formatting change) ← Older edit		Revision as of 18:31, 4 July 2013 (view source) Nathan wailes (talk \| contribs) Newer edit →
Line 9:		Line 9:

	[[Category:Intermediate Algebra Problems]]		[[Category:Intermediate Algebra Problems]]
		+	{{MAA Notice}}

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
All AIME Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "1996 AIME Problems/Problem 3"

Revision as of 18:31, 4 July 2013

Problem

Solution

See also