Discussions of these problems have already started on AoPS. Let's start a few over here. :)
Let and be different points on a circle such that is not a diameter. Let be the tangent line to at . Point is such that is the midpoint of the line segment . Point is chosen on the shorter arc of so that the circumcircle of triangle intersects at two distinct points. Let be the common point of and that is closer to . Line meets again at . Prove that the line is tangent to .
An integer is given. A collection of soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove players from this row leaving a new row of players in which the following conditions hold:
() no one stands between the two tallest players,
() no one stands between the third and fourth tallest players,
() no one stands between the two shortest players.
Show that this is always possible.
An ordered pair of integers is a primitive point if the greatest common divisor of and is . Given a finite set of primitive points, prove that there exist a positive integer and integers such that, for each in , we have: