Školjka

%V0 We consider the following system with $q=2p$: $$$\begin{align*} a_{11}x_{1}+\ldots&+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots&+a_{2q}x_{q}=0,\\ &\vdots \\ a_{p1}x_{1}+\ldots&+a_{pq}x_{q}=0,\\ \end{align*}$$$ 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 $j=1, \ldots ,q$.