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HMMT 十一月 2015 · GEN 赛 · 第 5 题

HMMT November 2015 — GEN Round — Problem 5

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Let S be a subset of the set { 1 , 2 , 3 , . . . , 2015 } such that for any two elements a, b ∈ S , the differencea − b does not divide the sum a + b . Find the maximum possible size of S .
解析

英文解析

  1. Let S be a subset of the set { 1 , 2 , 3 , . . . , 2015 } such that for any two elements a, b ∈ S , the differencea − b does not divide the sum a + b . Find the maximum possible size of S .
    Proposed by: Sam Korsky
    Answer: 672
    From each of the sets { 1 , 2 , 3 } , { 4 , 5 , 6 } , { 7 , 8 , 9 } , . . . at most 1 element can be in S . This leads to an
    ⌈ ⌉
    2015
    upper bound of = 672 which we can obtain with the set { 1 , 4 , 7 , . . . , 2014 } . 3