IrMO 2006

Are there integers $x,y$ and $z$ which satisfy the equation $$z^2=(x+1)(y^2-1)+n$$ when (a) $n=2006$ (b) $n=2007$ ?

$P$ and $Q$ are points on the equal sides $AB$ and $AC$ respectively of an isosceles triangle $ABC$ such that $AP=CQ$ . Moreover, neither $P$ nor $Q$ is a vertex of $ABC$ . Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of the triangle $ABC$ .

Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$ , $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$

The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building.

$\ast$ $\ast$ room with locked external doorstarting position

$ABC$ is a triangle with points $D$ , $E$ on $BC$ with $D$ nearer $B$ ; $F$ , $G$ on $AC$ , with $F$ nearer $C$ ; $H$ , $K$ on $AB$ , with $H$ nearer $A$ . Suppose that $AH=AG=1$ , $BK=BD=2$ , $CE=CF=4$ , $\angle B=60^\circ$ and that $D$ , $E$ , $F$ , $G$ , $H$ and $K$ all lie on a circle. Find the radius of the incircle of triangle $ABC$ .