IMO 1976 Problem 5

We consider the following system with $q=2p$ :

$$\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0, a_{21}x_{1}+\ldots+a_{2q}x_{q}=0, \ldots , a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0, \end{matrix}$$

in which every coefficient is an element from the set $\{-1,0,1\}$ $.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:a.) all $x_{j}, j=1,\ldots,q$ are integers $;$ b.) there exists at least one j for which $x_{j} \neq 0;$ c.) $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$