What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular \(n\)-gon?
a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?
b) The same problem, but for a regular polygon with 12 sides.
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.
a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.
b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.
It is known that a camera located at \(O\) cannot see the objects \(A\) and \(B\), where the angle \(AOB\) is greater than \(179^\circ\). 1000 such cameras are placed in a Cartesian plane. All of the cameras simultaneously take a picture. Prove that there will be a picture taken in which no more than 998 cameras are visible.
In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.
In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.
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.
a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?