HMMT 二月 2004 · GEN2 赛 · 第 6 题
HMMT February 2004 — GEN2 Round — Problem 6
题目详情
英文原题
- a and b are positive integers. When written in binary, a has 2004 1’s, and b has 2005 1’s
(not necessarily consecutive). What is the smallest number of 1’s a + b could possiblyhave?
解析
英文解析
- a and b are positive integers. When written in binary, a has 2004 1’s, and b has 2005 1’s
(not necessarily consecutive). What is the smallest number of 1’s a + b could possiblyhave?
Solution: 1
Consider the following addition:
111 · · · 100 · · · 01 + 11 · · · 11 = 1000 · · · · · · · · · 00
By making the blocks of 1’s and 0’s appropriately long, we can ensure that the addends 2
4008 2005
respectively contain 2004 and 2005 1’s. (To be precise, we get a = 2 − 2 + 1 and
2005
b = 2 − 1.) Then the sum has only one 1. And clearly it is not possible to get anyless than one 1.