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HMMT 十一月 2024 · THM 赛 · 第 4 题

HMMT November 2024 — THM Round — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Compute ⌊ ⌋ 4 4 4 ∑ ∑ ∑ ite . 5 i =1 t =1 e =1
解析
  1. Compute 4 4 4 X X X ite . 5 i =1 t =1 e =1 Proposed by: Derek Liu Answer: 168 Solution: Note that 5 never divides ite because 5 is prime. Thus, we can pair up the terms: since ite ≡ − (5 − i ) te mod 5, ite (5 − i ) te ite (5 − i ) te
  • = + − 1 . 5 5 5 5 Thus ! 4 4 4 4 4 4 3 X X X X X X ite ite 4 = − 5 5 2 i =1 t =1 e =1 i =1 t =1 e =1 ! 3 4 X 1 64 = i − 5 2 i =1 1 3 = (10) − 32 5 = 2 · 100 − 32 = 168 .