Question 76
Given are $n+1$ real linear equations in $n$ variables (of the form $a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}=a$). Prove that each $=$ sign can be replaced with either $\leq$ or $\geq$ so that the resulting $n+1$ inequalities have the following property: for every choice of real numbers $x_{1}, x_{2}, \ldots, x_{n}$, at least one inequality is true.