31st Vietnamese Mathematical Olympiad 1993 Problems

31st Vietnamese Mathematical Olympiad 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, a_n+1 = 2a_nb_n/(a_n + b_n), b_n+1 = √(a_n+1b_n). Show that the two sequences converge to the same limit, and find the limit.