2005 AIME II Problems/Problem 13
Problem
Let be a polynomial with integer coefficients that satisfies and Given that has two distinct integer solutions and find the product
Solution
Solution 1
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 and the other is and their product is .
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 | ||
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