1. Suppose that mutually distinct real numbers $a, b, c, d$ satisfy the inequalities

$$
a b+c d>b c+a d>a c+b d
$$

If $a$ is the largest of these four numbers, which one of them is the smallest? (Josef Tkadlec)
Solution. The inequality between the first two expressions will be rearranged by subtracting the right side and subsequently factoring out:

$$
\begin{aligned}
a b+c d-b c-a d & >0 \\
a(b-d)-c(b-d) & >0 \\
(a-c)(b-d) & >0
\end{aligned}
$$

Since $a>c$, the second parenthesis must also be positive, thus $b>d$.

Similarly, we will rearrange the inequality $b c+a d>a c+b d$, resulting in

$$
\begin{aligned}
b c+a d-a c-b d & >0 \\
b(c-d)-a(c-d) & >0 \\
(b-a)(c-d) & >0
\end{aligned}
$$

The first parenthesis is negative due to $a>b$, and therefore the second one must also be negative. Hence, we conclude that $d>c$.

We have thus derived the chain of inequalities $a>b>d>c$, from which it is clear that the smallest of the four numbers $a, b, c, d$ can only be $c$.

Note 1. Since the performed rearrangements were equivalent, we can state that both inequalities in the problem are satisfied whenever the real numbers $a, b, c, d$ satisfy $a>b>d>c$.

Note 2. By comparing the first expression with the last one, the inequality $b>c$ can be proven analogously. If this inequality is proven together with $d>c$, it is sufficient for a complete solution. However, if only $b>c$ and $b>d$ are proven, it is still possible that $c>d$.

$\triangleright$ For a complete solution, award 6 points, with three points for the proof of each inequality $b>d$ and $d>c$ (or $b>c$ and $d>c$). If only the pair of inequalities $b>c$ and $b>d$ is proven, award 4 points.

$\triangleright$ For finding one of the product forms $(a-c)(b-d)>0, (b-a)(c-d)>0, (a-d)(b-c)>0$, award 2 points. In the case of finding multiple product forms, award 4 points if the relevant inequalities lead to a complete solution (see the note at the end of the solution), and 3 points otherwise.

$\triangleright$ The existence of a suitable quadruple of numbers is already guaranteed in the problem statement, so it is not necessary to provide an example. For the same reason, it is not necessary to proceed equivalently in any rearrangements or to mention any equivalence.