2005 AIME II Problems/Problem 13
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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 • Resources) | ||
| Preceded by Problem 12 | Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
