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.
Prove that a convex quadrilateral \(ABCD\) can be drawn inside a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
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.
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.
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.