# Problem A. 663. (February 2016)

**A. 663.** There are given two positive integers: \(\displaystyle k\) and \(\displaystyle \ell\). A square with horizontal and vertical sides is divided into finitely many rectangles by line segments such that the following statements are satisfied: \(\displaystyle (i)\) every horizontal or vertical line of the plane contains at most one of the segments; \(\displaystyle (ii)\) no two segments cross each other in their interiors; \(\displaystyle (iii)\) every horizontal line, intersecting the square but not containing any of the segments, intersects exactly \(\displaystyle k\) rectangles; \(\displaystyle (iv)\) every vertical line, intersecting the square but not containing any of the segments, intersects exactly \(\displaystyle \ell\) rectangles. What can be the number of rectangles?

*Russian problem*

(5 pont)

**Deadline expired on March 10, 2016.**

### Statistics:

8 students sent a solution. 5 points: Baran Zsuzsanna, Gáspár Attila, Glasznova Maja, Imolay András, Williams Kada. 1 point: 1 student. 0 point: 2 students.

Problems in Mathematics of KöMaL, February 2016