31st Vietnam 1993 problems

A1.  f : [√1995, ∞) → R is defined by f(x) = x(1993 + √(1995 - x2) ). 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 n1945 for all n.
B1.  The tetrahedron ABCD has its vertices on the fixed sphere S. Find all configurations which minimise AB2 + AC2 + AD2 - BC2 - BD2 - CD2.
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 a0, a1, a2, ... and b0, b1, b2, ... by a0 = 2, b0 = 1, an+1 = 2anbn/(an + bn), bn+1 = √(an+1bn). Show that the two sequences converge to the same limit, and find the limit.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
22 July 2002