HMMT 二月 2018 · ALGNT 赛 · 第 9 题

HMMT February 2018 — ALGNT Round — Problem 9

Assume the quartic x − ax + bx − ax + d = 0 has four real roots ≤ x , x , x , x ≤ 2. Find the 1 2 3 4 2 ( x + x )( x + x ) x 1 2 1 3 4 maximum possible value of (over all valid choices of a, b, d ). ( x + x )( x + x ) x 4 2 4 3 1

解析

Assume the quartic x − ax + bx − ax + d = 0 has four real roots ≤ x , x , x , x ≤ 2. Find the 1 2 3 4 2 ( x + x )( x + x ) x 1 2 1 3 4 maximum possible value of (over all valid choices of a, b, d ). ( x + x )( x + x ) x 4 2 4 3 1 Proposed by: Allen Liu 5 Answer: 4 We can rewrite the expression as 2 x ( x + x )( x + x )( x + x )( x + x ) 1 1 1 2 1 3 1 4 4 · 2 x ( x + x )( x + x )( x + x )( x + x ) 4 1 4 2 4 3 4 4 1 2 x f ( − x ) 1 4 · 2 x f ( − x ) 4 1 where f ( x ) is the quartic. We attempt to find a simple expression for f ( − x ). We know that 1 3 f ( − x ) − f ( x ) = 2 a · x + 2 a · x 1 1 1 1 Since x is a root, we have 1 3 f ( − x ) = 2 a · x + 2 a · x 1 1 1 Plugging this into our previous expression: 2 3 x x + x 1 4 1 · 2 3 x x + x 4 1 4 1 x + 1 x 1 1 x + 4 x 4 1 1 The expression x + is maximized at x = 2 , and minimized at x = 1. We can therefore maximize x 2 5 the numerator with x = 2 and minimize the denominator with x = 1 to achieve the answer of . It 1 4 4 √ 10 − 1 can be confirmed that such an answer can be achieved such as with x = x = . 2 3 3