KURSCHAK 1963 Problem 1

$mn$ students all have different heights. They are arranged in $m > 1$ rows of $n > 1$ . In each row select the shortest student and let $A$ be the height of the tallest such. In each column select the tallest student and let $B$ be the height of the shortest such. Which of the following are possible: $A < B$ , $A = B$ , $A > B$ ? If a relation is possible, can it always be realized by a suitable arrangement of the students?