PUMaC 2017 · 团队赛 · 第 2 题

PUMaC 2017 — Team Round — Problem 2

(3) Let a % b denote the remainder when a is divided by b . Find (100% i ). i =1

解析

Note that 100 = b c x + 100% x . Therefore, if < x ≤ , then 100% x = 100 − nx . x n +1 n Hence, we can split the initial sum into several smaller sums as follows: 100 21 100 ∑ ∑ ∑ (100% i ) = (100% i ) + (100% i ) i =1 1 21 ( ) 21 25 33 50 100 ∑ ∑ ∑ ∑ ∑ = (100% i ) + 80 · 100 − 4 i + 3 i + 2 i + 1 i 1 i =21 i =26 i =34 i =51 21 ∑ = (100% i ) + 8000 − (4 · 115 + 3 · 236 + 2 · 714 + 1 · 3775) 1 21 ∑ = (100% i ) + 1629 . 1 The sum from 1 to 21 can be evaluated directly to be 72, which makes the final answer 1701 . Problem written by Nathan Bergman