Difference between revisions of "2005 AIME II Problems/Problem 13"
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[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Revision as of 18:20, 22 March 2007
Problem
Let be a polynomial with integer coefficients that satisfies
and
Given that
has two distinct integer solutions
and
find the product
Solution
Define the polynomial . By the givens,
,
,
and
. Note that for any polynomial
with integer coefficients and any integers
we have
divides
. So
divides
, and so
must be one of the eight numbers
and so
must be one of the numbers
or
. Similarly,
must divide
, so
must be one of the eight numbers
or
. Thus,
must be either 19 or 22. Since
obeys the same conditions and
and
are different, one of them is 19 and the other is 22 and their product 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 |