KURSCHAK 1963

$mn$ 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?

$A$ is an acute angle. Show that $$\left(1 +\frac{1}{sen A}\right)\left(1 +\frac{1}{cos A}\right)> 5$$

A triangle has no angle greater than $90^o$ . Show that the sum of the medians is greater than four times the circumradius.