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HMMT 二月 2000 · ORAL 赛 · 第 2 题

HMMT February 2000 — ORAL Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [30] How many positive solutions are there to x + 7 x + 14 x + 1729 x − 1379 x = 0 ? How many positive integer solutions ? n n n
解析
  1. Dividing by x , we get the equation x + 7 x + 14 x + 1729 x − 1379 = 0. This is a monotonically increasing function of x on [0 , ∞ ). From f (0) < 0 and f ( x ) → ∞ as x → ∞ we see that there’s only one positive solution. In fact, f (1) > 0, so there’s no positive integer solution. n n n − 1 n − 1 n − 1