4. Let be an acute triangle with orthocenter , and let be a point on the side , between and . The points and are the feet of the altitudes drawn from and , respectively. is the circumcircle of triangle , and is a point such that is a diameter of . Similarly, is the circumcircle of triangle , and is a point such that is a diameter of . Show that the points , and are collinear.
5. Let be the set of all rational numbers greater than zero. Let be a function satisfying the following conditions
(i) for all ,
(ii) for all ,
(iii) There exists a rational number such that .
Show that for all .
6. Let be an integer, and consider a circle with equally spaced points marked on it. Consider all labellings of these points with the numbers such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels with , the chord joining the points labelled and does not intersect the chord joining the points labelled and .
Let be the number of beautiful labellings and let be the number of ordered pairs of positive integers such that and . Prove that