SE-SKOLORNAS 1999

Solve $|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30$ .

Circle $C$ center $O$ touches externally circle $C'$ center $O'$ . A line touches $C$ at $A$ and $C'$ at $B$ . $P$ is the midpoint of $AB$ . Show that $\angle OPO' = 90^o$ .

Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$ .

An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$ . What are the possible values of $x$ ?

$x_i$ are non-negative reals. $x_1 + x_2 + ...+ x_n = s$ . Show that $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n \le \frac{s^2}{4}$ .

$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$ . Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.