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