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第54届国际数学奥林匹克(IMO2013)第2天试题  

2013-07-25 06:14:14|  分类: 奥赛 |  标签: |举报 |字号 订阅

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Day 2 - 24 July 2013

4. Let ABC be an acute triangle with orthocenter H , and let W be a point 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. \omega_1 is the circumcircle of triangle BWN , and X is a point such that WX is a diameter of \omega_1 . Similarly, \omega_2 is the circumcircle of triangle CWM , and Y is a point such that WY is a diameter of \omega_2 . Show that the points X, Y , and H are collinear.
 
5. Let \mathbb Q_{>0} be the set of all rational numbers greater than zero. Let f: \mathbb Q_{>0} \to \mathbb R be a function satisfying the following conditions:

(i) f(x)f(y) \geq f(xy) for all x, y \in \mathbb Q_{>0} ,
(ii) f(x+y) \geq f(x) + f(y) for all x, y \in \mathbb Q_{>0} ,
(iii) There exists a rational number 第54届国际数学奥林匹克(IMO2013)第2天试题 - sqing55 - sqing55 such that f (a) = a .

Show that f(x) = x for all x \in \mathbb Q_{>0} .
 
6. Let n\geq 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,\dots, 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\leq n and \gcd(x,y)=1 . Prove that M=N+1.
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