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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 points)

Deadline expired on 10 March 2016.

Statistics on problem A. 663.
 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

•  Támogatóink: Morgan Stanley