Question 1
Prove that for any positive integer $m$ there exists a positive integer $n$, such that any $n$ different points on a plane can be partitioned into $m$ non-empty sets, *convex hulls* of which would share a common point.
*Convex hull* of a finite set $X$ of points on a plane is a set of points that lie inside or on the border of at least one convex polygon with vertices in $X$, including degenerate ones, i.e. a segment and a point are considered to be convex polygons. No three vertices of a convex polygon are collinear. A polygon contains its border.
*Convex hull* of a finite set $X$ of points on a plane is a set of points that lie inside or on the border of at least one convex polygon with vertices in $X$, including degenerate ones, i.e. a segment and a point are considered to be convex polygons. No three vertices of a convex polygon are collinear. A polygon contains its border.