EE-MO 2003

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$ -gon and $A_1B_2...B_n$ is a regular $n$ -gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$ . Find $\angle B_2A_1A_2$ .

Find all positive integers $n$ such that $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$

In the rectangle $ABCD$ with $|AB|<2 |AD|$ , let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$ . Prove that $FAD$ is an isosceles triangle.