This problem is from AMM 10737 (1999-05) proposed by Hassan Ali Shah Ali,Tehran,Iran.
Let $m$ and $n$ be postive integers with $n\ge 2m$, and let $a_{1}\le a_{2}\le\cdots\le a_{n}$ be postive integers such that $$a_{n}<m+\dfrac{1}{2m}\sum_{i=1}^{m}\binom{n}{2i}\binom{2i}{i}$$
show that there exist two different $n-$ tuples $(\xi_{1},\xi_{2},\cdots,\xi_{n})$ and $(\delta_{1},\delta_{2},\cdots,\delta_{n})$,with entries $0,1,2$ ,such that $\sum_{j=1}^{n}\xi_{j}=\sum_{j=1}^{n}\delta_{j}\le 2m$ and $\sum_{j=1}^{n}\xi_{i}a_{j}=\sum_{j=1}^{n}\delta_{j}a_{j}$
I searched the magazine for a long time, and went through the answers from 2000 to 2003 without finding that the question had been answered. Of course I could have missed it. This seems like an interesting question, so I'm going to turn to you. Thank you
