4th Iberoamerican 1989 problems

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A1.  Find all real solutions to: x + y - z = -1; x2 - y2 + z2 = 1, -x3 + y3 + z3 = -1.
A2.  Given positive real numbers x, y, z each less than π/2, show that π/2 + 2 sin x cos y + 2 sin y cos z > sin 2x + sin 2y + sin 2z.
A3.  If a, b, c, are the sides of a triangle, show that (a - b)/(a + b) + (b - c)/(b + c) + (c - a)/(a + c) < 1/16.
B1.  The incircle of the triangle ABC touches AC at M and BC at N and has center O. AO meets MN at P and BO meets MN at Q. Show that MP·OA = BC·OQ.
B2.  The function f on the positive integers satisfies f(1) = 1, f(2n + 1) = f(2n) + 1 and f(2n) = 3 f(n). Find the set of all m such that m = f(n) for some n.
B3.  Show that there are infinitely many solutions in positive integers to 2a2 - 3a + 1= 3b2 + b.

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
11 July 2002