CWMO 2001

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) $A_1 \cup A_2 \cup \cdots \cup A_n = A$ , (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 $b < a \leq \frac {4}{3}b$ .