A square is cut by 18 straight lines, 9 of which are parallel to one side of the square and the other 9 parallel to the other – perpendicular to the first 9 – dividing the square into 100 rectangles. It turns out that exactly 9 of these rectangles are squares. Prove that among these 9 squares there will be two that are identical.
Which rectangles with whole sides are there more of: with perimeter of 1996 or with perimeter of 1998? (The rectangles \(a \times b\) and \(b \times a\) are assumed to be the same).
9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).