HMMT 二月 2013 · 代数 · 第 4 题
HMMT February 2013 — Algebra — Problem 4
题目详情
英文原题
Determine all real values of Afor which there exist distinct complex numbers x1, x2such that the
following three equations hold:
x1(x1+ 1) = A
x2(x2+ 1) = A
x4
1+ 3x3
1+ 5x1=x4
2+ 3x3
2+ 5x2.
解析
英文解析
- 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) .
Thus, in order for the last equation to hold, we need ( A + 7) x = ( A + 7) x , from which it follows that 1
1 2
A = − 7. These steps are reversible, so A = − 7 indeed satisfies the needed condition.
3 2