Problem

Source: Polish Math Olympiad 2021 2nd round p2 day 2

Tags: geometry, rectangle, combinatorial geometry



Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_j$ have a common point for all $i, j \in \{1,..., n\}$, $i \ne j$. Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices