SMO 2004

Let $m,n$ be integers so that $m \ge n > 1$ . Let $F_1,...,F_k$ be a collection of $n$ -element subsets of $\{1,...,m\}$ so that $F_i\cap F_j$ contains at most $1$ element, $1 \le i < j \le k$ . Show that $k\le \frac{m(m-1)}{n(n-1)}$

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$ , such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$ . Justify your answer.

Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$ . A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$ . Let $E$ be a point on the segment $AD$ , different from $A$ and $D$ . The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$ . The line $BE$ intersects $\Gamma_2$ at $M$ and $N$ . (i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$ . (ii) If the centre of $\Gamma_3$ is $O$ , prove that $OD$ is perpendicular to $BC$ .

If $0 <x_1,x_2,...,x_n\le 1$ , where $n \ge 1$ , show that $$\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1$$