▪ Japan Mathematical Olympiad Finals 2013

1. Let be positive integers with . There are persons, each person belongs to exactly one of group , group , group and more than or equal to one person belong to any groups. Show that sweets can be delivered to persons in such way that all of the following condition are satisfied. 

At least one sweet are delivered to each person.

sweet are delivered to each person belonging to group

If , then  

2. Find all functions such that the equality 

holds for all

3. Let be a positive integer. Find the minimum value of positive integer for which there exist positive integers such that:

are all square numbers.

4. Given an acute-angled triangle ABC, let be the orthocenter. A cirlcle passing through the points and a cirlcle with a diameter intersect at two distinct points . Let be the foot of the perpendicular drawn from to line , and let be the foot of the perpendicular drawn from to line . Show that . 

5. Let be a positive integer. Given are points of which any three points are not collinear. For , rotating half-line clockwise in about the pivot gives half-line Find the maximum value of the number of the pairs of such that line segments and intersect at except endpoints. 

Note that : and

Πηγή: artofproblemsolving