Problem

Source: 2012cmo,problem6

Tags: modular arithmetic, pigeonhole principle, inequalities, combinatorics proposed, combinatorics



Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers. Proposed by Huawei Zhu