Problem

Source: (2021-) 2022 XV 15th Dürer r Math Competition Finals Day 1 E5 E+2

Tags: geometry, rectangle, combinatorics, areas, combinatorial geometry



Annie drew a rectangle and partitioned it into $n$ rows and $k$ columns with horizontal and vertical lines. Annie knows the area of the resulting $n \cdot k$ little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given $n$ and $k$ at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the $n \cdot k$ little rectangles? For example in the case $n = 3$ and $k = 4$ revealing the areas of the $10$ small rectangles if enough information to find the areas of the remaining two little rectangles.