PAMO 2007

Find all natural numbers $N$ consisting of exactly $1112$ digits (in decimal notation) such that: (a) The sum of the digits of $N$ is divisible by $2000$ ; (b) The sum of the digits of $N+1$ is divisible by $2000$ ; (c) $1$ is a digit of $N$ .

Let $A$ , $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$ ), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$ . Find the locus of $C'$ .

In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$ . (Not all towns need to be visited.)

Solve the following system of equations for real $x,y$ and $z$ : \begin{eqnarray*} x &=& \sqrt{2y+3} y &=& \sqrt{2z+3} z &=& \sqrt{2x+3}. \end{eqnarray*}

For which positive integers $n$ is $231^n-222^n-8 ^n -1$ divisible by $2007$ ?

An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?