A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon $($see the drawing$)$. Is it possible to reconstruct the original square using just this information?
What is the minimum number of points necessary to mark inside a convex $n$-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?
A convex figure and point A inside it are given. Prove that there is a chord $($that is, a segment joining two boundary points of a convex figure$)$ passing through point A and dividing it in half at point A.
Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside the quadrilateral.
We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.