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?
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.
a) Prove that the axes of symmetry of a regular polygon intersect at one point.
b) Prove that the regular \(2n\)-gon has a centre of symmetry.
a) The convex \(n\)-gon is cut by diagonals that do not cross to form triangles. Prove that the number of these triangles is equal to \(n - 2\).
b) Prove that the sum of the angles at the vertices of a convex \(n\)-gon is \((n - 2) \times 180^{\circ}\).
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.