Difference between revisions of "1996 AIME Problems/Problem 3"
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== Solution == | == Solution == | ||
| − | Using [[Simon's Favorite Factoring Trick]], rewrite as <math>[(x+7)(y-3)]^n = (x+7)^n(y-3)^n</math>. Both [[binomial expansion]]s will contain <math>n+1</math> non-like terms; their product will contain <math>(n+1)^2</math> terms, as each term will have an unique power of <math>x</math> or <math>y</math> and so none of the terms will need to be collected. Hence <math>(n+1)^2 | + | Using [[Simon's Favorite Factoring Trick]], we rewrite as <math>[(x+7)(y-3)]^n = (x+7)^n(y-3)^n</math>. Both [[binomial expansion]]s will contain <math>n+1</math> non-like terms; their product will contain <math>(n+1)^2</math> terms, as each term will have an unique power of <math>x</math> or <math>y</math> and so none of the terms will need to be collected. Hence <math>(n+1)^2 \ge 1996</math>, the smallest square after <math>1996</math> is <math>2025 = 45^2</math>, so our answer is <math>45 - 1 = \boxed{044}</math>. |
== See also == | == See also == | ||
Revision as of 18:27, 2 June 2012
Problem
Find the smallest positive integer 
for which the expansion of 
, after like terms have been collected, has at least 1996 terms.
Solution
Using Simon's Favorite Factoring Trick, we rewrite as ![$[(x+7)(y-3)]^n = (x+7)^n(y-3)^n$](http://latex.artofproblemsolving.com/5/7/d/57dfb01ed05f0de2e7ded3ce81ade5442ac9fa66.png)
. Both binomial expansions will contain 
non-like terms; their product will contain 
terms, as each term will have an unique power of 
or 
and so none of the terms will need to be collected. Hence 
, the smallest square after 
is 
, so our answer is 
.
See also
| 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 | ||
