Japan Mathematical Olympiad Finals 2013

1. Let n,\ k be positive integers with n\geq k. There are n persons, each person belongs to exactly one of group 1, group 2,\ \cdots, group k and more than or equal to one person belong to any  groups. Show that n^2 sweets can be delivered to n persons in such way that all of  the following condition are satisfied.

a/. At least one sweet are delivered to each person.

b/. a_i sweet are delivered to each person belonging to group i\ (1\leq i\leq k).

c/. If 1\leq i<j\leq k, then a_i>a_j.

2. Find all functions f:\mathbb{Z}\rightarrow\mathbb{R} such that  the equality

f(m)+f(n)=f(mn)+f(m+n+mn) holds for all m,n\in\mathbb{Z}.

3. Let n\geq 2 be a positive integer. Find the minimum value of positive integer m for which there exist positive integers a_1,\ a_2,\ \cdots, a_n such that :

a/. \ a_1<a_2<\cdots <a_n=m

b/. \displaystyle\frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2} are all square numbers.

4. Given an acute-angled triangle ABC, let H be the orthocenter. A cirlcle passing through the points B,\ C and a cirlcle with a diameter AH intersect at two distinct points X,\ Y. Let D be the foot of the perpendicular drawn from A to line BC, and let K be the foot of the perpendicular drawn from D to line XY. Show that \angle{BKD}=\angle{CKD}.

5. Let n be a positive integer. Given are points P_1,\ P_2,\ \cdots,\ P_{4n} of which any three points are not collinear. For i=1,\ 2,\ \cdots,\ 4n, rotating half-line P_iP_{i-1} clockwise in 90^\circ about the pivot  P_i gives half-line P_iP_{i+1}. Find the maximum value of the number of the pairs of (i,\ j) such that line segments P_iP_{i+1} and P_jP_{j+1} intersect at except endpoints.

Note that : P_0=P_{4n},\ P_{4n+1}=P_1 and 1\leq i<j\leq 4n.

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