HMMT 二月 2004 · 冲刺赛 · 第 25 题
HMMT February 2004 — Guts Round — Problem 25
题目详情
- [9] Suppose x − ax + bx − 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 ?
解析
- Suppose x − ax + bx − 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 ? 6 C X A B Solution: 3 / 2 We know pqr = 48 since the product of the roots of a cubic is the constant term. Now, √ 1 2 3 6 3 3
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- ≥ 3 = p q r pqr 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.