HMMT 二月 2004 · 冲刺赛 · 第 8 题

HMMT February 2004 — Guts Round — Problem 8

[6] I have chosen five of the numbers { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product?

解析

I have chosen five of the numbers { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product? Solution: 420 Giving you the product of the five numbers is equivalent to telling you the product of the two numbers I didn’t choose. The only possible products that are achieved by more than one pair of numbers are 12 ( { 3 , 4 } and { 2 , 6 } ) and 6 ( { 1 , 6 } and { 2 , 3 } ). But in the second case, you at least know that the two unchosen numbers have odd sum (and so the five chosen numbers have odd sum also). Therefore, the first case must hold, and the product of the five chosen numbers is 1 · 2 · 5 · 6 · 7 = 1 · 3 · 4 · 5 · 7 = 420 .