JMO 1992 Problem 5

Suppose that $n\geq 2$ be integer and $a_1,\ a_2,\ a_3,\ a_4$ satisfy the following condition:

i) $\ n$ and $a_i\ (i = 1,\ 2,\ 3,\ 4)$ are relatively prime.

ii) $\ (ka_1)_n + (ka_2)_n + (ka_3)_n + (ka_4)_n = 2n$ for $k = 1,\ 2,\ \cdots ,\ n - 1$.

Note that $(a)_n$ expresses the divisor when $a$ is divided by $n$.

Prove that $(a_1)_n,\ (a_2)_n,\ (a_3)_n,\ (a_4)_n$ can be divided into two pair with sum $n$.