2013 IMO Problems
Prove that for any pair of positive integers and , there exist positive integers (not necessarily different) such that
A configuration of points in the plane is called Colombian if it consists of red points and blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:
- no line passes through any point of the configuration;
- no region contains points of both colours.
Find the least value of such that for any Colombian configuration of points, there is a good arrangement of lines.
Let the excircle of triangle opposite the vertex be tangent to the side at the point . Define the points on and on analogously, using the excircles opposite and , respectively. Suppose that the circumcentre of triangle lies on the circumcircle of triangle . Prove that triangle is right-angled.
Let be an acute triangle with orthocenter , and let be a point on the side , lying strictly between and . The points and are the feet of the altitudes from and , respectively. Denote by is [sic] the circumcircle of , and let be the point on such that is a diameter of . Analogously, denote by the circumcircle of triangle , and let be the point such that is a diameter of . Prove that and are collinear.
Let be the set of all positive rational numbers. Let be a function satisfying the following three conditions:
(i) for all , we have ; (ii) for all , we have ; (iii) there exists a rational number such that .
Prove that for all .
Let be an integer, and consider a circle with equally spaced points marked on it. Consider all labellings of these points with the numbers 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 with , the chord joining the points labelled and does not intersect the chord joining the points labelled and .
Let be the number of beautiful labelings, and let N be the number of ordered pairs of positive integers such that and . Prove that
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