CZ-SK-MO 2007

A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations: (a) move the stone to a square that shares a common side with the square in which it stands;(b) move it to a square sharing only one common vertex with the square in which it stands.

In addition, we are required that the first step must be (b). Find all integers $n$ such that the stone can go through a certain path visiting every square exactly once.

In a cyclic quadrilateral $ABCD$ , let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$ .Prove that triangle $LMR$ is isosceles.

Consider a function $f:\mathbb N\rightarrow \mathbb N$ such that for any two positive integers $x,y$ , the equation $f(xf(y))=yf(x)$ holds. Find the smallest possible value of $f(2007)$ .

The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$ , then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$ , are in $M$ . Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$ ?

In an acute-angled triangle $ABC$ ( $AC\ne BC$ ), let $D$ and $E$ be points on $BC$ and $AC$ , respectively, such that the points $A,B,D,E$ are concyclic and $AD$ intersects $BE$ at $P$ . Knowing that $CP\bot AB$ , prove that $P$ is the orthocenter of triangle $ABC$ .

Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$ . (It means, those three fractions make a permutation of $x, y$ , and $z$ .)