IT-CESENATICO 2003

Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$ .

A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$ . Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.

Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$ . Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$ . A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.

There are two sorts of people on an island: knights, who always talk truth, and scoundrels, who always lie. One day, the people establish a council consisting of $2003$ members. They sit around a round table, and during the council each member said: "Both my neighbors are scoundrels". In a later day, the council meets again, but one member could not come due to illness, so only $2002$ members were present. They sit around the round table, and everybody said: "Both my neighbors belong to the sort different from mine". Is the absent member a knight or a scoundrel?

In each lattice-point of an $m \times n$ grid and in the centre of each of the formed unit squares a pawn is placed.

a) Find all such grids with exactly $500$ pawns. b) Prove that there are infinitely many positive integers $k$ for which therer is no grid containing exactly $k$ pawns.

Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$ . The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there. Find all $n$ for which he can select the guests in such an order to serve all the guests.