HMMT 二月 2018 · 冲刺赛 · 第 8 题

HMMT February 2018 — Guts Round — Problem 8

[ 6 ] Suppose a real number x > 1 satisfies log (log x ) + log (log x ) + log (log x ) = 0 . 2 4 4 16 16 2 Compute log (log x ) + log (log x ) + log (log x ) . 2 16 16 4 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2018, February 10, 2018 — GUTS ROUND Organization Team Team ID#

解析

[ 6 ] Suppose a real number x > 1 satisfies log (log x ) + log (log x ) + log (log x ) = 0 . 2 4 4 16 16 2 Compute log (log x ) + log (log x ) + log (log x ) . 2 16 16 4 4 2 Proposed by: Michael Tang 1 Answer: − 4 Let A and B be these sums, respectively. Then ( ) ( ) ( ) log x log x log x 16 2 4 B − A = log + log + log 2 4 16 log x log x log x 4 16 2 = log (log 4) + log (log 16) + log (log 2) 2 16 4 2 16 4 ( ) ( ) 1 1 = log + log 4 + log 2 4 16 2 2 ( ) 1 = ( − 1) + 1 + − 4 1 = − . 4 1 Since A = 0, we have the answer B = − . 4