HMMT 十一月 2014 · 冲刺赛 · 第 12 题

HMMT November 2014 — Guts Round — Problem 12

[ 8 ] Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and − signs before each of the integers, yielding an expression +1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 11 + 12 − · · · − 199. What is the value of the resulting expression? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND Organization Team Team ID# 25

解析

[ 8 ] Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and − signs before each of the integers, yielding an expression +1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 11 + 12 − · · · − 199. What is the value of the resulting expression? Answer: − 100 Group the numbers into (1 − 2 + 3 − 4 + ... + 18 − 19) + (21 − 22 + ... + 38 − 39) + ... + (181 − 182 + ... + 198 − 199). We can easily show that each group is equal to − 10, and so the answer is − 100. 25