HMMT 二月 2022 · 冲刺赛 · 第 17 题

HMMT February 2022 — Guts Round — Problem 17

[11] Compute the number of positive real numbers x that satisfy 16 ⌊ log x ⌋ 13 2 3 · 2 − x = 2022 x .

解析

[11] Compute the number of positive real numbers x that satisfy 16 ⌊ log x ⌋ 13 2 3 · 2 − x = 2022 x . Proposed by: Daniel Zhu Answer: 9 ⌊ log x ⌋ i i +1 i 2 Solution: Let f ( x ) = 3 · 2 − x . Note that for each integer i , if x ∈ [2 , 2 ) , then f ( x ) = 3 · 2 − x . i i +1 i +1 i This is a line segment from (2 , 2 ) to (2 , 2 ) , including the first endpoint but not the second. 16 13 Now consider the function f ( x ) /x . This consists of segments of decreasing functions connecting i 3 i +16 i +1 3 i − 13 3 i − 13 10 (2 , 2 ) and (2 , 2 ). Note that for each − 1 ≤ i ≤ 7 , we have that 2 ≤ 2 < 2022 < 11 3 i +16 2 ≤ 2 . This gives us 9 solutions in total.