25th Vietnamese Mathematical Olympiad 1987 Problems

25th Vietnamese Mathematical Olympiad 1987 problems

A1.  Let x_n = (n+1)π/3974. Find the sum of all cos(± x₁ ± x₂ ± ... ± x₁₉₈₇).
A2.  The sequences a₀, a₁, a₂, ... and b₀, b₁, b₂, ... are defined as follows. a₀ = 365, a_n+1 = a_n(a_n¹⁹⁸⁶ + 1) + 1622, b₀ = 16, b_n+1 = b_n(b_n³ + 1) - 1952. Show that there is no number in both sequences.

A3.  There are n > 2 lines in the plane, no two parallel. The lines are not all concurrent. Show that there is a point on just two lines.

B1.  x₁, x₂, ... , x_n are positive reals with sum X and n > 1. h ≤ k are two positive integers. H = 2^h and K = 2^k. Show that x₁^K/(X - x₁)^H-1 + x₂^K/(X - x₂)^H-1 + x₃^K/(X - x₃)^H-1 + ... + x_n^K/(X - x_n)^H-1 ≥ X^K-H+1/( (n-1)^2H-1n^K-H). When does equality hold?

B2.  The function f(x) is defined and differentiable on the non-negative reals. It satisfies | f(x) | ≤ 5, f(x) f '(x) ≥ sin x for all x. Show that it tends to a limit as x tends to infinity.

B3.  Given 5 rays in space from the same point, show that we can always find two with an angle between them of at most 90^o.