SMO 2003

A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$ . Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$ ·

Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

For any given prime $p$ , determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$ . Justify your answer

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$ . If the longest side from $S$ is $SA$ , prove that $BS > CS$ .