### 24th Vietnam 1986 problems A1.  a1, a2, ... , an are real numbers such that 1/2 ≤ ai ≤ 5 for each i. The real numbers x1, x2, ... , xn satisfy 4xi2 - 4aixi + (ai - 1)2 = 0. Let S = (x1 + x2 + ... + xn)/n, S2 = (x12 + x22 + ... + xn2)/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 + ... + xn2) ≥ xn(x1 + x2 + ... + xn-1) for all real xi. 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. To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
jscholes@kalva.demon.co.uk
23 July 2002