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HMMT 二月 2012 · 冲刺赛 · 第 3 题

HMMT February 2012 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 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 .
解析
  1. [ 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.