A quadrilateral is given; \(A\), \(B\), \(C\), \(D\) are the successive midpoints of its sides, \(P\) and \(Q\) are the midpoints of its diagonals. Prove that the triangle \(BCP\) is equal to the triangle \(ADQ\).
In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
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.