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PUMaC 2021 · 代数（B 组） · 第 1 题

PUMaC 2021 — Algebra (Division B) — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Let x, y be distinct positive real numbers satisfying 1 1 x √ √ √ √ p
  • = . 3 x + y − x − y x + y + x − y y √ x a + b If = for positive integers a, b, c with gcd( a, c ) = 1, find a + b + c . y c y
解析

1 . 5 Kris still got the right answer. Given that x > 10 , determine the largest possible value of y. Proposed by: Frank Lu Answer: 4 Let u = log x. Then, we know from logarithm properties that we’re looking for yu, and Kris 10 1 y y − 1 computed u instead. For these to be equal, we have that u = y . We’re thus looking for 1 y − 1 y − 1 the largest y such that y > 1 . 1 , or that y > 1 . 5 . Trying some small values of y, note 2 3 that y = 5 yields that 5 < (2 . 25) = 5 . 0625 , but 4 > 1 . 5 = 3 . 375 , so our desired answer is 4 .