1996 AIME Problems/Problem 3
Contents
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 . 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 .
Alternatively, when , the exponents of or in can be any integer between and inclusive. Thus, when , there are terms and, when , there are terms. Therefore, we need to find the smallest perfect square that is greater than . From trial and error, we get and . Thus, .
Solution FASTER
the floor of the square root of 1996 easily 44
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 |
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