$AB$ is a chord (not a diameter) of a circle with centre $O$. Let $T$ be a point on segment $OB$. The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto $AB$ . Prove that $AS \cdot BC = TE \cdot TD$.

Find all positive integers $m$ and $n$ such that $n^m - m$ divides $m^2 + 2m$.

Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$.

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.

(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$. (ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.