# 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, 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

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