OPM 2009

A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$ , $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?

Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$ , respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$ .

Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?

Circumferences $C_1$ and $C_2$ have different radios and are externally tangent on point $T$ . Consider points $A$ on $C_1$ and $B$ on $C_2$ , both different from $T$ , such that $\angle BTA=90^{\circ}$ . What is the locus of the midpoints of line segments $AB$ constructed that way?