The sequence $ \{x_n\}$ satisfies $ x_1 = \frac {1}{2}, x_{n + 1} = x_n + \frac {x_n^2}{n^2}$. Prove that $ x_{2001} < 1001$.

$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value, a) Prove that $ AB \geq 2BC$, b) Find the value of $ AQ \cdot BQ$.

Let $ n, m$ be positive integers of different parity, and $ n > m$. Find all integers $ x$ such that $ \frac {x^{2^n} - 1}{x^{2^m} - 1}$ is a perfect square.

Let $ x, y, z$ be real numbers such that $ x + y + z \geq xyz$. Find the smallest possible value of $ \frac {x^2 + y^2 + z^2}{xyz}$.

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor = 4x + 3$.

$ P$ is a point on the exterior of a circle centered at $ O$. The tangents to the circle from $ P$ touch the circle at $ A$ and $ B$. Let $ Q$ be the point of intersection of $ PO$ and $ AB$. Let $ CD$ be any chord of the circle passing through $ Q$. Prove that $ \triangle PAB$ and $ \triangle PCD$ have the same incentre.

Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 - \sin 2x)\sin (x + \frac {\pi}{4}) = 1$.

We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if (i) $ A_1 \cap A_2 \cap \cdots \cap A_n = A$, (ii) $ A_i \cap A_j \neq \emptyset$. Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A = \{1, 2, \ldots, m\}$, there exists a set $ A_i$ ($ 1 \leq i \leq 14$) such that there are two elements $ a, b$ of $ A_i$ such that $ b < a \leq \frac {4}{3}b$.