2010 AIME I Problems/Problem 5
Contents
Problem
Positive integers ,
,
, and
satisfy
,
, and
. Find the number of possible values of
.
Solution
Using the difference of squares, , where equality must hold so
and
. Then we see
is maximal and
is minimal, so the answer is
.
Solution 2
Since must be greater than
, it follows that the only possible value for
is
(otherwise the quantity
would be greater than
). Therefore the only possible ordered pairs for
are
,
, ... ,
, so
has
possible values.
See also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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All AIME Problems and Solutions |