## Task 2 - 300732

200 students are arranged in the form of a rectangle, with 20 students in each longitudinal row and 10 students in each transverse row.

Now, from each transverse row, the smallest student is called out. Among the 20 students thus identified, the tallest one is denoted as $A$. The 20 students then return to their original positions.

Next, from each longitudinal row, the tallest student is called out, and among the 10 students thus identified, the smallest one is denoted as $B$. It turns out that $B$ has a different height from $A$.

Investigate which of the two students, $A$ and $B$, must be the taller one under these conditions!
There are exactly the following three possibilities:

(1) $A$ and $B$ are in the same column.

In this case, $B$ is greater than $A$, since $B$ is as large as possible in the mentioned column and of a different size than $A$.

(2) $A$ and $B$ are in the same row.

In this case, $B$ is also greater than $A$, since $A$ is as small as possible in the mentioned row and of a different size than $B$.

(3) $A$ and $B$ are neither in the same column nor in the same row.

Let $C$ be the student who is in the same row as $A$ and in the same column as $B$. For this student, it holds that: $B$ is greater than $C$ or equally large as $C$, and $C$ is greater than $A$ or equally large as $A$. Since $B$ is not equally large as $A$, it follows again: $B$ is greater than $A$.

Thus, in every case: The larger of the two students $A$ and $B$ must be $B$.