Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

Suppose $x^{3}-a x^{2}+b x-48$ is a polynomial with three positive roots $p, q$ , and $r$ such that $p<q<r$ . What is the minimum possible value of $1 / p+2 / q+3 / r$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We know $p q r=48$ since the product of the roots of a cubic is the constant term. Now, $\frac{1}{p}+\frac{2}{q}+\frac{3}{r} \geq 3 \sqrt[3]{\frac{6}{p q r}}=\frac{3}{2}$ by AM-GM, with equality when $1 / p=2 / q=3 / r$ . This occurs when $p=2, q=4$ , $r=6$ , so $3 / 2$ is in fact the minimum possible value.

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