※ 引述《FAlin (FA（バルシェ應援）)》之銘言：
: 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.