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HMMT 二月 2004 · 冲刺赛 · 第 10 题

HMMT February 2004 — Guts Round — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [7] Let f ( x ) = x + x + x + x + · · · , for all real x such that the sum converges. Forhow many real numbers x does f ( x ) = x ?

英文原题

What is the largest possible value of x + y ?

解析

英文解析

  1. Let f ( x ) = x + x + x + x + · · · , for all real x such that the sum converges. For howmany real numbers x does f ( x ) = x ?
    Solution: 2
    2 2 2
    Clearly x = 0 works. Otherwise, we want x = x / (1 − x ), or x + x − 1 = 0. Discard
    √
    the negative root (since the sum doesn’t converge there), but ( − 1 + 5) / 2 works, for a total of 2 values.