【IMO 2013 讨论帖】

题目来源http://www.artofproblemsolving.com/Forum/viewforum.php?f=951
翻译由楼主本人完成，由于外语水平不高所以翻译不足之处还望各位指出

IMO 2013 Problem 1 Prove that for any two positive integers there exist positive integers such that

证明对任意正整数k,n存在正整数m1,m2,...,mk使得上式成立

IMO 2013 Problem 2 Given red and blue points in the plane, no three of them on a line. We aim to split the plane by lines (not passing through these points) into regions such that there are no regions containing points of both the colors. What is the least number of lines that always suffice?
平面上有2013个红点和2014个蓝点，任意三点不共线，我们用不过这些点的直线将平面划分成若干个区域，使得所有区域中的点都是同色的，求满足此划分的直线的条数的最小值

IMO 2013 Problem 3 Let be a triangle and let , , and be points of contact of the excircles with the sides , , and , respectively. Prove that if the circumcenter of lies on the circumcircle of , then is a right triangle.
三角形ABC中，A1,B1，C1分别是边BC,AC,AB与各自旁切圆的切点，如果三角形A1B1C1的外心在三角形ABC的外接圆上，证明三角形ABC是直角三角形

IMO 2013 Problem 4
锐角三角形ABC中H是垂心，W在边BC上，M和N分别是三角形过B,C的高的垂足，WX是三角形BWN的外接圆的直径，WY是三角形CWM的外接圆的直径，求证X,Y,H三点共线

IMO 2013 Problem 5 Let be the set of all rational numbers greater than zero. Let be a function satisfying the following conditions:
(i) for all ,
(ii) for all ,
(iii) There exists a rational number such that .
Show that for all .
f是一个从正有理数到实数的映射且满足以上条件，求证f(x)=x

第三题仍在公关中……
第四题，设两圆另一交点为T，S为A在BC上的射影，由AN*AB=AM*AC知A,T,S共线且AT*AW=AH*AS,得H,S,Y共线，同理X,H,S共线，又由相交两圆的内接三角形的性质知X,S,T共线，所以X,H,S,T共线
第五题仿照cauchy方程即可
第六题组合渣渣表示还是等大神吧……

我擦楼上打错好多，【A,T,S共线】→【A,T,W】共线
【H,S,Y共线】、【H,S,X共线】→【H,T,Y共线】、【H,T,X共线】
【X,H,S,T共线】→【X,H,T,Y共线】