HMMT 二月 2001 · ADV 赛 · 第 9 题
HMMT February 2001 — ADV Round — Problem 9
题目详情
英文原题
- Find the real solutions of (2 x + 1)(3 x + 1)(5 x + 1)(30 x + 1) = 10.
2 2
解析
英文解析
- Find the real solutions of (2 x + 1)(3 x + 1)(5 x + 1)(30 x + 1) = 10.
Solution: (2 x + 1)(3 x + 1)(5 x + 1)(30 x + 1) = [(2 x + 1)(30 x + 1)][(3 x + 1)(5 x + 1)] =
2 2 2
(60 x + 32 x + 1)(15 x + 8 x + 1) = (4 y + 1)( y + 1) = 10, where y = 15 x + 8 x . The quadratic
√
9 − 4 ± 31
equation in y yields y = 1 and y = − . For y = 1, we have 15 x + 8 x − 1 = 0, so x = .2
4 15
9 9
For y = − , we have 15 x + 8 x + , which yields only complex solutions for x . Thus the real 2
4 4
√
− 4 ± 31
solutions are .
2 215