2015 IMO Problems
We say that a finite set in the plane is balanced if, for any two different points , in , there is a point in such that . We say that is centre-free if for any three points , , in , there is no point in such that .
- Show that for all integers , there exists a balanced set consisting of points.
- Determine all integers for which there exists a balanced centre-free set consisting of points.
Determine all triples of positive integers such that each of the numbers is a power of 2.
(A power of 2 is an integer of the form where is a non-negative integer ).
Let be an acute triangle with . Let be its circumcircle, its orthocenter, and the foot of the altitude from . Let be the midpoint of . Let be the point on such that . Assume that the points , , , , and are all different, and lie on in this order.
Prove that the circumcircles of triangles and are tangent to each other.
Triangle has circumcircle and circumcenter . A circle with center intersects the segment at points and , such that , , , and are all different and lie on line in this order. Let and be the points of intersection of and , such that , , , , and lie on in this order. Let be the second point of intersection of the circumcircle of triangle and the segment . Let be the second point of intersection of the circumcircle of triangle and the segment .
Suppose that the lines and are different and intersect at the point . Prove that lies on the line .
Let be the set of real numbers. Determine all functions : satisfying the equation
for all real numbers and .
The sequence of integers satisfies the conditions:
(i) for all ,
(ii) for all .
Prove that there exist two positive integers and for whichfor all integers and such that .