AMC 2010 12A P21

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Problem Statement

The polynomial $x^3-ax^2+bx-2010$ has three positive integer roots. What is the smallest possible value of $a$?

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Solution

$$f(x) = x^3-ax^2+bx-2010 = (x - r_1)(x - r_2)(x - r_3), \; r_1, r_2, r_3 \in \mathbb{N}$$ $$f(x) = x^3 - (r_1 + r_2 + r_3)x^2 + (r_1r_2 + r_2r_3 + r_1r_3)x - r_1r_2r_3$$ $$\implies r_1r_2r_3 = 2010, min(a) = min(r_1 + r_2 + r_3)$$ $$2010 = 2 \times 3 \times 5 \times 67$$ $$min(r_1 + r_2 + r_3) \text{ is when } r_1, r_2, r_3 \text { are as close as possible}$$ $$\text{WLOG let } r_1 = 67, r_2 = 6, r_3 = 5 \implies min(a) = min(r_1 + r_2 + r_3) = 67 + 6 + 5 = 78$$