Erdős-Ginzburg-Ziv theorem

If $a_{1},a_{2},\ldots,a_{2n-1}$ is a set of integers, then there exists a subset $a_{i_{1}},a_{i_{2}},\ldots,a_{i_{n}}$ of $n$ integers such that

a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}}\equiv 0\pmod{n}.

The theorem is also known as the EGZ theorem.

References

1 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.
2 Hao,P. On a Congruence modulo a Prime Amer. Math. Monthly, vol. 113, (2006), 652-654