HMMT 二月 2014 · 冲刺赛 · 第 8 题

英文解析

8. [ 5 ] The numbers 2 , 2 , · · · , 2 , 2 = 65536 are written on a blackboard. You repeatedly take twonumbers on the blackboard, subtract one from the other, erase them both, and write the result of thesubtraction on the blackboard. What is the largest possible number that can remain on the blackboardwhen there is only one number left?
   Answer: 131069 If we reverse the order of the numbers in the final subtraction we perform,
   then the final number will be negated. Thus, the possible final numbers come in pairs with oppositesigns. Therefore, the largest possible number is the negative of the smallest possible number. To getthe smallest possible number, clearly we can take the smallest number originally on the board andsubtract all of the other numbers from it (you can make this rigorous pretty easily if needed), so the
   ∑
   k 16
   smallest possible number is 1 − 2 = 1 − 131070 = − 131069, and thus the largest possible numberk =1
   is 131069.