Difference between revisions of "2004 AIME I Problems/Problem 7"
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Revision as of 20:01, 4 July 2013
Problem
Let be the coefficient of
in the expansion of the product
Find
Contents
[hide]Solution
Solution 1
Let our polynomial be .
It is clear that the coefficient of in
is
, so
, where
is some polynomial divisible by
.
Then and so
, where
is some polynomial divisible by
.
However, we also know
.
Equating coefficients, we have , so
and
.
Solution 2
Let be the set of integers
. The coefficient of
in the expansion is equal to the sum of the product of each pair of distinct terms, or
. Also, we know that
where the left-hand sum can be computed from:

and the right-hand sum comes from the formula for the sum of the first perfect squares. Therefore,
.
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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