4. Let ABC be an acute triangle with orthocenter H, and let W be apoint on
the side BC, between B and C. The points M and N are the feet of the
altitudes drawn from B and C, respectively. ω_1 is the circumcircle of
triangle BWN, and X is a point such that WX is a diameter of ω_1.
Similarly, ω_2 is the circumcircle of triangle CWM, and Y is a point
such that WY is a diameter of ω_2. show that the points X, Y, and H are
collinear.
5. Let Q>0 be the set of all rational numbers greater than zero. Let
f: Q>0 → R be a function satisfying the following conditions:
(i) f(x)f(y) ≧ f(xy) for all x,y ∈ Q>0,
(ii) f(x+y) ≧ f(x) + f(y) for all x,y ∈ Q>0
(iii) There exists a rational number a>1 such that f(a) = a
Show that f(x) = x for all x∈Q>0.
6. Let n≧3 be an integer, and consider a circle with n+1 equally spaced
points marked on it. Consider all labellings of these points with the
numbers 0,1,..., n 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 a<b<c<d with a+d=b+c, the chord joining the
points labelled a and d does not intersect the chord joining the points
labelled b and c.
Let M be the number of beautiful labellings and let N be the number of
ordered pairs (x,y) of positive integers such that x+y≦n and
gcd(x,y)=1.
Prove that M = N+1.