HMMT 二月 2013 · 代数 · 第 5 题
HMMT February 2013 — Algebra — Problem 5
题目详情
英文原题
Let aand bbe real numbers, and let r, s, and tbe the roots of f(x) = x3+ax2+bx−1. Also, g(x) =x3+mx2+nx+phas roots r2,s2, and t2. Ifg(−1) = −5, find the maximum possible value ofb.
解析
英文解析
- Let a and b be real numbers, and let r, s, and t be the roots of f ( x ) = x + ax + bx − 1. Also,
3 2 2 2 2
g ( x ) = x + mx + nx + p has roots r , s , and t . If g ( − 1) = − 5, find the maximum possible valueof b .
√
2 2 2 2 2 2 2 2 2 2
Answer: 1 + 5 By Vieta’s Formulae, m = − ( r + s + t ) = − a + 2 b , n = r s + s t + t r =
2 2 2 2 2
b + 2 a , and p = − 1. Therefore, g ( − 1) = − 1 − a + 2 b − b − 2 a − 1 = − 5 ⇔ ( a + 1) + ( b − 1) = 5.
This is an equation of a circle, so b reaches its maximum when a + 1 = 0 ⇒ a = − 1. When a = − 1,
√ √
b = 1 ± 5, so the maximum is 1 + 5.
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