2007/08 AHSMC Part 1 Problem 10

Problem 10 of 16

The number of pairs $(a, b)$ of positive integers such that all three roots of the cubic equation $x^3 - 10x^2 + ax - b = 0$ are positive integers is

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Solution

Let the positive integral roots be $r \le s \le t$ . Then $x^3 - 10x^2 + ax - b = (x-r)(x-s)(x-t)$ . Expansion yields $x^3 - (r+s+t)x^2 + (st + tr + rs)x - rst$ . Hence $r + s + t = 10$ . The possible partitions are $(1,1,8)$ , $(1,2,7)$ , $(1,3,6)$ , $(1,4,5)$ , $(2,2,6)$ , $(2,3,5)$ , $(2,4,4)$ and $(3,3,4)$ . The answer is (B).

University of Alberta

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