HMMT 二月 2014 · 冲刺赛 · 第 8 题

HMMT February 2014 — Guts Round — Problem 8

[ 5 ] The numbers 2 , 2 , · · · , 2 , 2 = 65536 are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT 2014, 22 FEBRUARY 2014 — GUTS ROUND Organization Team Team ID#

解析

[ 5 ] The numbers 2 , 2 , · · · , 2 , 2 = 65536 are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when 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 opposite signs. Therefore, the largest possible number is the negative of the smallest possible number. To get the smallest possible number, clearly we can take the smallest number originally on the board and subtract all of the other numbers from it (you can make this rigorous pretty easily if needed), so the ∑ 16 k smallest possible number is 1 − 2 = 1 − 131070 = − 131069, and thus the largest possible number k =1 is 131069.