IMO Shortlist 1991 problem 12
Dodao/la:
arhiva2. travnja 2012. Given any integer
![n \geq 2,](/media/m/b/3/c/b3c5a25a999394108439f699d794082e.png)
assume that the integers
![a_1, a_2, \ldots, a_n](/media/m/9/2/c/92c14c25a50ea2e6e7d3f457e8ea9a16.png)
are not divisible by
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and, moreover, that
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
does not divide
![\sum^n_{i=1} a_i.](/media/m/0/e/c/0ec11efa10657dd47a110d473eb06988.png)
Prove that there exist at least
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
different sequences
![(e_1, e_2, \ldots, e_n)](/media/m/e/d/4/ed4e5624413f6913301589a2315088af.png)
consisting of zeros or ones such
![\sum^n_{i=1} e_i \cdot a_i](/media/m/1/9/0/190f3567f08019de07cc87d5667a5cbc.png)
is divisible by
%V0
Given any integer $n \geq 2,$ assume that the integers $a_1, a_2, \ldots, a_n$ are not divisible by $n$ and, moreover, that $n$ does not divide $\sum^n_{i=1} a_i.$ Prove that there exist at least $n$ different sequences $(e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $\sum^n_{i=1} e_i \cdot a_i$ is divisible by $n.$
Izvor: Međunarodna matematička olimpijada, shortlist 1991