[問題] IMO 2013 in Colombia Day 2

[FAlin (FA（バルシェ應援）)]

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.

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

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◆ From: 124.11.128.7
推 Dawsen:有請學長分析難度
推 JGU:5. 證到恆正, 遞增. 後面不會  XD
推 cmrafsts:4沒難度5正常1小時6MARKING SHIT裱人
→ cmrafsts:17分是可以拿到的
→ Dawsen:我覺得2,3,5比1,4更直接. 6想不出來 如果金牌線>35再重想XD
推 Dawsen:JGU學長是不是還有一個條件沒用？
推 cmrafsts:如果有恆正遞增那應該也同時做出，f(整數)>=自己，就快了
推 hahaj6u4503:6的給分標準一向很摳@@ 感覺今年23兩題是關鍵，第二題
→ hahaj6u4503:感覺會卡掉一些人，金牌線可能又不高了
推 cmrafsts:我看目前協調分數感覺會是15~16-22~23-30~31
推 Dawsen:看目前的部份分數 台灣可能有3~4金!
推 LimSinE:JGU可得1分