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HMMT 十一月 2022 · 冲刺赛 · 第 14 题

HMMT November 2022 — Guts Round — Problem 14

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [9] Real numbers x and y satisfy the following equations: y − 1 x = log (10 + 1) − 1 10 x y = log (10 + 1) − 1 . 10 x − y Compute 10 .
解析
  1. [9] Real numbers x and y satisfy the following equations: y − 1 x = log (10 + 1) − 1 10 x y = log (10 + 1) − 1 . 10 x − y Compute 10 . Proposed by: Ankit Bisain 101 Answer: 110 Solution: Taking 10 to the power of both sides in each equation, these equations become: x y − 1 − 1 10 = 10 + 1 · 10 y x − 1 10 = (10 + 1) · 10 . x y Let a = 10 and b = 10 . Our equations become: 10 a = b/ 10 + 1 10 b = a + 1 , and we are asked to compute a/b . Subtracting the equations gives 10 a − 10 b = b/ 10 − a = ⇒ 11 a = 101 b/ 10 , giving an answer of 101 / 110.