How prove this there exsit set $B$ such $card(B)>\dfrac{n}{3}$,


Let $a_{i}\in N^{+}$, and the set $A=\{a_{1},a_{2},\cdots,a_{n}\}$,

show that:

There exists a set $B\subset A$, such that $card(B)>\dfrac{n}{3}$, 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 sets $A=\{1,2,3,\cdots,2n,2n+1\}$, and the sets $B$ such $B\subset A$, and for any $x,y\in B$, then $x+y\notin B$,

Find the $|B|_{max}$

this problem answer is $|B|_{max}=n+1$. The full solution (Mathematical induction
) can see china BBs:

But for this I can’t prove it, Thank you very much

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