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HMMT 二月 2019 · 冲刺赛 · 第 3 题

HMMT February 2019 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 3 ] Let S ( x ) denote the sum of the digits of a positive integer x. Find the maximum possible value of S ( x + 2019) − S ( x ) .
解析
  1. [ 3 ] Let S ( x ) denote the sum of the digits of a positive integer x. Find the maximum possible value of S ( x + 2019) − S ( x ) . Proposed by: Alec Sun Answer: 12 We note that S ( a + b ) ≤ S ( a ) + S ( b ) for all positive a and b , since carrying over will only decrease the b sum of digits. (A bit more rigorously, one can show that S ( x + a · 10 ) − S ( x ) ≤ a for 0 ≤ a ≤ 9 . ) Hence we have S ( x + 2019) − S ( x ) ≤ S (2019) = 12, and equality can be achieved with x = 100000 for example.