Difference between revisions of "2004 AIME I Problems/Problem 12"
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Revision as of 16:21, 27 April 2008
Problem
Let be the set of ordered pairs
such that
and
and
are both even. Given that the area of the graph of
is
where
and
are relatively prime positive integers, find
The notation
denotes the greatest integer that is less than or equal to
Solution
is even when
Likewise:
is even when
Graphing this creates a series of rectangles which become smaller as you move toward the origin.
The interval of each box is given by the sequence
The interval is given by
Each box is the product of one term of each sequence. The sum of these boxes is simply the product of the sum of each sequence or:
Geometric sums are taken:
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |