HMMT 二月 2021 · 冲刺赛 · 第 20 题

HMMT February 2021 — Guts Round — Problem 20

[12] Let f ( x ) = x − 3 x . Compute the number of positive divisors of ⌊ ( ( ( ( ( ( ( ( ))))))))⌋ 5 f f f f f f f f , 2 where f is applied 8 times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT Spring 2021, March 06, 2021 — GUTS ROUND Organization Team Team ID#

解析

[12] Let f ( x ) = x − 3 x . Compute the number of positive divisors of ⌊ ( ( ( ( ( ( ( ( ))))))))⌋ 5 f f f f f f f f , 2 where f is applied 8 times. Proposed by: Akash Das Answer: 6562 1 1 1 1 1 1 3 3 3 Solution: Note that f ( y + ) = ( y + ) − 3( y + ) = y + . Thus, f (2 + ) = 2 + , and in 3 3 y y y y 2 2 ( ) k k 1 3 1 general f 2 + = 2 + , where f is applied k times. It follows that we just need to find the k 3 2 2 ⌊ ⌋ 8 8 3 1 3 8 number of divisors of 2 + = 2 , which is just 3 + 1 = 6562. 8 3 2