HMMT 二月 2012 · 冲刺赛 · 第 3 题
HMMT February 2012 — Guts Round — Problem 3
题目详情
英文原题
[2] Suppose xandyare real numbers such that −1< x < y < 1. Let Gbe the sum of the geometric series whose first term is xand whose ratio is y, and let G′be the sum of the geometric series whose first term is yand ratio is x. IfG=G′, find x+y.
解析
英文解析
- [ 2 ] Suppose x and y are real numbers such that − 1 < x < y < 1. Let G be the sum of the geometric
′
series whose first term is x and whose ratio is y , and let G be the sum of the geometric series whose
′
first term is y and ratio is x . If G = G , find x + y .
′
Answer: 1 We note that G = x/ (1 − y ) and G = y/ (1 − x ). Setting them equal gives x/ (1 − y ) =
2 2
y/ (1 − x ) ⇒ x − x = y − x ⇒ ( x + y − 1)( x − y ) = 0, so we get that x + y − 1 = 0 ⇒ x + y = 1.