Question:
Let $a_{i}\in\mathbb N^{+}$, and the set $A=\{a_{1},a_{2},\dots,a_{n}\}$. Show that there exists a set $B\subset A$, such that $\mathrm{card}(B)>\dfrac{n}{3}$, and that for any $x,y\in B$, then $x+y\notin B$.
This problem is from China Nankai university math competition today. How prove this? Thank you.
Maybe this problem is an old problem, but I can't find it.
At last, I find a similar problem: let the set $A=\{1,2,3,\dots,2n,2n+1\}$, and the sets $B$ such $B\subset A$, and for any $x,y\in B$, then $x+y\notin B$. Find $\max |B|$.
The answer is $\max |B|=n+1$. For a full solution (mathematical induction) see BBs:http://zhidao.baidu.com/question/260607678.html
But for this I can't prove it. Thank you very much.