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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