China Western Mathematical Olympiad

Suppose $k<4$. Take $a$, $b$, $c$, $d\in [k,\sqrt{4k}]$. Then for any permutation $p$, $q$, $r$, $s$ of $a$, $b$, $c$, $d$, consider the equation $x^2+px+q=0$, its discriminant $$\Delta =p^2-4q<4k-4q\le 4k-4k=0.$$ Therefore it has no real roots. So $k\ge 4$.

Suppose $4\le a<b<c<d$. Consider the following equations: $$x^2+dx+a=0,$$ $$x^2+cx+b=0.$$ Observe that their discriminants $$\Delta =d^2-4a>4(d-a)>0$$ and $$\Delta =c^2-4b>4(c-b)>0.$$ Then the above two equations have two distinct real roots.

Suppose these two equations have the same real root $\beta$. Then we have $${\beta }^2+d\beta +a=0,$$ $${\beta }^2+c\beta +b=0.$$ Taking their difference yields $\beta =\frac{b-a}{d-c}>0$. Then ${\beta }^2+d\beta +a>0$, which leads to a contradiction. So $k=4$.