IMO Shortlist 2005 problem C7
Dodao/la:
arhiva2. travnja 2012. Suppose that
![a_1](/media/m/6/1/7/6173ac27c63013385bea9def9ff2b61e.png)
,
![a_2](/media/m/4/0/1/401f4cdfec59fba73ae32fa6769c72cb.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
are integers such that
![n\mid a_1 + a_2 + \ldots + a_n](/media/m/6/3/5/635cfc068587f25bc0c2e8d10aa8d8bd.png)
.
Prove that there exist two permutations
![\left(b_1,b_2,\ldots,b_n\right)](/media/m/1/8/e/18e87b372a02823d4d5b7f7d8f62820a.png)
and
![\left(c_1,c_2,\ldots,c_n\right)](/media/m/6/0/f/60fbb526896f8561b280969fff15a685.png)
of
![\left(1,2,\ldots,n\right)](/media/m/3/1/b/31bdcf8eeb5db9417b1774d52ae8323b.png)
such that for each integer
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
with
![1\leq i\leq n](/media/m/1/e/e/1ee4f10bd28ea7bb5a88120fbf9e78a1.png)
, we have
%V0
Suppose that $a_1$, $a_2$, $\ldots$, $a_n$ are integers such that $n\mid a_1 + a_2 + \ldots + a_n$.
Prove that there exist two permutations $\left(b_1,b_2,\ldots,b_n\right)$ and $\left(c_1,c_2,\ldots,c_n\right)$ of $\left(1,2,\ldots,n\right)$ such that for each integer $i$ with $1\leq i\leq n$, we have
$$n\mid a_i - b_i - c_i$$
Izvor: Međunarodna matematička olimpijada, shortlist 2005