HMMT 二月 2003 · GEN2 赛 · 第 2 题

HMMT February 2003 — GEN2 Round — Problem 2

You have a list of real numbers, whose sum is 40. If you replace every number x on the list by 1 − x , the sum of the new numbers will be 20. If instead you had replaced every number x by 1 + x , what would the sum then be?

解析

You have a list of real numbers, whose sum is 40. If you replace every number x on the list by 1 − x , the sum of the new numbers will be 20. If instead you had replaced every number x by 1 + x , what would the sum then be? Solution: 100 Let n be the number of numbers on the list. If each initial number is replaced by its negative, the sum will then be − 40, and adding 1 to every number on this list increases the sum by n , so n − 40 = 20 ⇒ n = 60. Then, if we had simply added 1 to each of the initial numbers (without negating first), the sum would increase to 40 + n = 40 + 60 = 100.