HMMT 二月 2018 · ALGNT 赛 · 第 2 题
HMMT February 2018 — ALGNT Round — Problem 2
题目详情
- Compute the positive real number x satisfying 6 (2 x ) x = 3 . ⌊ ⌋ 2 n
解析
- Compute the positive real number x satisfying 6 (2 x ) x = 3 . Proposed by: Henrik Boecken √ 6 Answer: 3 6 2 t 6 t t 3 Let t = x , so x = 3. Taking this to the third power gives x = 27, or equivalently t = 3 . We can t see by inspection that t = 3, and this is the only solution as for t > 1, the function t is monotonically √ 6 t 6 increasing, and if 0 < t < 1, t < 1. Solving for x gives x = 3, or x = 3. ⌊ ⌋ 2 n