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. -- valuable sheaves 4 FELIDS ╔╦╦═╦╗╔═╦╦╦═╗ storyteller Blessing Card ║║║═║║║╚╣╩║═╣ JESTER ║║║║║╚╬╗║║║═╣ REVOLT ╚═╩╩╩═╩═╩╩╩═╝ PLAY THE JOKER AFFLICT / Fragment --

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