A1. f : [√1995, ∞) → R is defined by f(x) = x(1993 + √(1995 - x^{2}) ). Find its maximum and minimum values.
A2. 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.
A3. Find a function f(n) on the positive integers with positive integer values such that f( f(n) ) = 1993 n^{1945} for all n.
B1. 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}.
B2. 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.
B3. 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_{n}b_{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. |