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HMMT 二月 2000 · 团队赛 · 第 3 题

HMMT February 2000 — Team Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Find the sum of all in tegers from 1 to 1000 in lusiv e whi h on tain at least one 7 in their digits, i.e. nd 7 + 17 + ::: + 979 + 987 + 997.
解析
  1. The sum of the n um b ers from 700 to 799 is = 74950. The sum of 2 2 79 80 69 70 the n um b ers from 70 to 79 is = 745. So, all n um b ers that end from 2 2 70 to 79 (ex luding those starting with 7, sin e w e oun ted those from 700 to 799) is 745 9 + 10(100 + 200 + : : : + 600 + 800 + 900 ) = 44705. The sum of all n um b ers ending in 7 is 9(7 + 17 + 27 + 37 + 47 + 57 + 67 + 87 + 97) + 9(100 + 200 + : : : + 600 + 800 + 900) = 38187. So, the total sum of n um b ers on taining a 7 is 74950 + 44705 + 38186 = 157842 .