2015 IMO Problems
Contents
Problem 1
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.
Problem 2
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 ).
Problem 3
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.
Problem 4
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 .
Problem 5
Let be the set of real numbers. Determine all functions : satisfying the equation
for all real numbers and .
Problem 6
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 .
Problem 1
Let be the set of integers. Determine all functions such that, for all integers and ,
Problem 2
In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .
Prove that points , and are concyclic.
Problem 3
A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
Problem 4
Find all pairs of positive integers such that
Problem 5
The Bank of Bath issues coins with an on one side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation:
If there are exactly coins showing , then he turns over the coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be , which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration , let be the number of operations before Harry stops. For example, and . Determine the average value of over all possible initial configurations .
Problem 6
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .
2015 IMO (Problems) • Resources | ||
Preceded by 2014 IMO Problems |
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