HMMT 二月 2004 · 冲刺赛 · 第 8 题

英文解析

8. I have chosen five of the numbers { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . If I told you what their productwas, that would not be enough information for you to figure out whether their sumwas even or odd. What is their product?
   Solution: 420
   Giving you the product of the five numbers is equivalent to telling you the productof the two numbers I didn’t choose. The only possible products that are achieved bymore than one pair of numbers are 12 ( { 3 , 4 } and { 2 , 6 } ) and 6 ( { 1 , 6 } and { 2 , 3 } ). Butin 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 musthold, and the product of the five chosen numbers is
   1 · 2 · 5 · 6 · 7 = 1 · 3 · 4 · 5 · 7 = 420 .