HMMT 二月 2013 · 代数 · 第 4 题

HMMT February 2013 — Algebra — Problem 4

Determine all real values of A for which there exist distinct complex numbers x , x such that the 1 2 following three equations hold: x ( x + 1) = A 1 1 x ( x + 1) = A 2 2 4 3 4 3 x + 3 x + 5 x = x + 3 x + 5 x . 1 2 1 1 2 2 3 2

解析

Determine all real values of A for which there exist distinct complex numbers x , x such that the 1 2 following three equations hold: x ( x + 1) = A 1 1 x ( x + 1) = A 2 2 4 3 4 3 x + 3 x + 5 x = x + 3 x + 5 x . 1 2 1 1 2 2 Answer: − 7 Applying polynomial division, 4 3 2 2 x + 3 x + 5 x = ( x + x − A )( x + 2 x + ( A − 2)) + ( A + 7) x + A ( A − 2) 1 1 1 1 1 1 1 1 = ( A + 7) x + A ( A − 2) . 1 Thus, in order for the last equation to hold, we need ( A + 7) x = ( A + 7) x , from which it follows that 1 2 A = − 7. These steps are reversible, so A = − 7 indeed satisfies the needed condition. 3 2