※ 引述《FAlin (FA（バルシェ应援）)》之铭言：
: 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.

wow... 厉害... 原来归纳法行得通！

我后来是用 "以0开头k结尾的直线BL恰Φ(k)个"直接构造

不过可以把k当成第二个 可以证明假设0紧接的下一个是k的话

1. 所有相差是k的一定得按照顺序排 而且a, a+k中间不能有

间隔，否则考虑第一个不满足条件的元素可以导致矛盾2. 接下来就是mod k剩余系的排列问题。 第一个已经是余0

假设下一个是余x的话，可以证明之后余a的一定得在余a+x之前

出现，所以(x,k)=1, 而且一种这种排列都是唯一，且是BL

多的1是k=1的情形。BTW, 把k当成第二个 意思是把0当第一个

对了..(1)我有疑问。重新调位置之后，还会平行吗？ 如果会

平行，好像就不用证了。如果不平行, 0,2k+1也有可能相邻

还是说你只是写sketch, 进一步分析可发现不为BL?

(3)的ai+al-aj也有可能会=kd for some integer d?

这是要模k后的结果阿，所以小于2k

我不太理解学长对(1)的疑问，不过我上面写的应该不足以完成(1)的证明。2k+1时我们直接考虑0---2k+1

不，这样好像证不出(1)，虽然要可以证出但我在凌晨证不出请学长们先看看怎么证...简哥唬烂我说这样能证...

我也打算用归纳法，(1)太强了没观察到!!

喔有了，反正只要再管0n相邻，拖(3)救援WWW

偷偷问一下楼主是?XD

TWN1，台湾最低分 我知道楼上是谁喔~~~