VMO 1993

f : [√1995, ∞) → R is defined by f(x) = x(1993 + √(1995 - $x^2$) ). Find its maximum and minimum values.

ABCD is a quadrilateral such that AB is not parallel to CD, and BC is not parallel to AD. Variable points P, Q, R, S are taken on AB, BC, CD, DA respectively so that PQRS is a parallelogram. Find the locus of its center.

Find a function f(n) on the positive integers with positive integer values such that f( f(n) ) = 1993 $n^{1945}$ for all n.

The tetrahedron ABCD has its vertices on the fixed sphere S. Find all configurations which minimise $AB^2$ + $AC^2$ + $AD^2$ - $BC^2$ - $BD^2$ - $CD^2$.

1993 points are arranged in a circle. At time 0 each point is arbitrarily labeled +1 or -1. At times n = 1, 2, 3, ... the vertices are relabeled. At time n a vertex is given the label +1 if its two neighbours had the same label at time n-1, and it is given the label -1 if its two neighbours had different labels at time n-1. Show that for some time n > 1 the labeling will be the same as at time 1.

Define the sequences $a_0$, $a_1$, $a_2$, ... and $b_0$, $b_1$, $b_2$, ... by $a_0$ = 2, $b_0$ = 1, $a_{n+1}$ = $2a_nb_n$/($a_n$ + $b_n$), $b_{n+1}$ = √($a_{n+1}b_n$). Show that the two sequences converge to the same limit, and find the limit.