We consider the following system with
![q=2p](/media/m/e/3/4/e348e9fc09e21b924c10efa7c906db18.png)
:
![\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*}](/media/m/1/a/2/1a2df3b2a7d1dc078ed81523f10224e0.png)
in which every coefficient is an element from the set
![.](/media/m/b/d/d/bdd5ec3ff70fef87f72128d28ab734d1.png)
Prove that there exists a solution
![x_{1}, \ldots, x_{q}](/media/m/d/d/e/dde1176352b3b999c90639253bf6d075.png)
for the system with the properties:
a.) all
![x_{j}, j=1,\ldots,q](/media/m/7/0/5/705016e5623e255d52037e360dbb58eb.png)
are integers;
b.) there exists at least one j for which
![x_{j} \neq 0](/media/m/3/3/0/330458d1da86abcfcdc3bd5bc16d07f2.png)
;
c.)
![|x_{j}| \leq q](/media/m/3/1/7/3176a0fa81061b7c96abbdf6cc83fd55.png)
for any
![j=1, \ldots ,q](/media/m/d/1/9/d1981ac6103fc39ade1bfbb61e5f0d64.png)
.
%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 $j=1, \ldots ,q$.