24th Vietnamese Mathematical Olympiad 1986 Problems

24th Vietnamese Mathematical Olympiad 1986 Problems

A1.  a₁, a₂, ... , a_n are real numbers such that 1/2 ≤ a_i ≤ 5 for each i. The real numbers x₁, x₂, ... , x_n satisfy 4x_i² - 4a_ix_i + (a_i - 1)² = 0. Let S = (x₁ + x₂ + ... + x_n)/n, S₂ = (x₁² + x₂² + ... + x_n²)/n. Show that √S₂ ≤ S + 1.

A2.  P is a pyramid whose base is a regular 1986-gon, and whose sloping sides are all equal. Its inradius is r and its circumradius is R. Show that R/r ≥ 1 + 1/cos(π/1986). Find the total area of the pyramid's faces when equality occurs.

A3.  The polynomial p(x) has degree n and p(1) = 2, p(2) = 4, p(3) = 8, ... , p(n+1) = 2ⁿ⁺¹. Find p(n+2).

B1.  ABCD is a square. ABM is an equilateral triangle in the plane perpendicular to ABCD. E is the midpoint of AB. O is the midpoint of CM. The variable point X on the side AB is a distance x from B. P is the foot of the perpendicular from M to the line CX. Find the locus of P. Find the maximum and minimum values of XO.

B2.  Find all n > 1 such that (x₁² + x₂² + ... + x_n²) ≥ x_n(x₁ + x₂ + ... + x_n-1) for all real x_i.

B3.  A sequence of positive integers is constructed as follows. The first term is 1. Then we take the next two even numbers: 2, 4. Then we take the next three odd numbers: 5, 7, 9. Then we take the next four even numbers: 10, 12, 14, 16. And so on. Find the nth term of the sequence.