HMMT 二月 2013 · 代数 · 第 5 题

HMMT February 2013 — Algebra — Problem 5

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 value of b .

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 value of 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. Algebra Test