Difference between revisions of "2005 AIME II Problems/Problem 13"
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=== Solution 2 === | === Solution 2 === |
Revision as of 22:45, 14 February 2016
Contents
[hide]Problem
Let be a polynomial with integer coefficients that satisfies
and
Given that
has two distinct integer solutions
and
find the product
Solution
Solution 2
As above, we define , noting that it has roots at
and
. Hence
. In particular, this means that
. Therefore,
satisfy
, where
,
, and
are integers. This cannot occur if
or
because the product
will either be too large or not be a divisor of
. We find that
and
are the only values that allow
to be a factor of
. Hence the answer is
.
See also
2005 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.