1976 IMO 第 5 题

(NET 3) ${ }^{\mathrm{IMO} 5}$ Let a set of $p$ equations be given, $$\begin{gathered} a_{11} x_{1}+\cdots+a_{1 q} x_{q}=0 \\ a_{21} x_{1}+\cdots+a_{2 q} x_{q}=0 \\ \vdots \\ a_{p 1} x_{1}+\cdots+a_{p q} x_{q}=0, \end{gathered}$$ with coefficients $a_{i j}$ satisfying $a_{i j}=-1,0$, or +1 for all $i=1, \ldots, p$ and $j=1, \ldots, q$. Prove that if $q=2 p$, there exists a solution $x_{1}, \ldots, x_{q}$ of this system such that all $x_{j}(j=1, \ldots, q)$ are integers satisfying $\left|x_{j}\right| \leq q$ and $x_{j} \neq 0$ for at least one value of $j$.

(NET 3) ${ }^{\mathrm{IMO} 5}$ 给出一组 $p$ 方程， $$\begin{gathered} a_{11} x_{1}+\cdots+a_{1 q} x_{q}=0 \\ a_{21} x_{1}+\cdots+a_{2 q} x_{q}=0 \\ \vdots \\ a_{p 1} x_{1}+\cdots+a_{p q} x_{q}=0，\end{gathered}$$，系数 $a_{i j}$ 满足 $a_{i j}=-1,0$，或对于所有 $i=1、\ldots、p$ 和 $j=1、\ldots、q$ 为 +1。证明如果$q=2 p$，则存在该系统的解$x_{1}, \ldots, x_{q}$，使得所有$x_{j}(j=1, \ldots, q)$都是满足$\left|x_{j}\right|的整数对于$j$的至少一个值，\leq q$ 和 $x_{j} \neq 0$。