39th Vietnam 2001 problems

A1.  A circle center O meets a circle center O' at A and B. The line TT' touches the first circle at T and the second at T'. The perpendiculars from T and T' meet the line OO' at S and S'. The ray AS meets the first circle again at R, and the ray AS' meets the second circle again at R'. Show that R, B and R' are collinear.
A2.  Let N = 6n, where n is a positive integer, and let M = aN + bN, where a and b are relatively prime integers greater than 1. M has at least two odd divisors greater than 1. Find the residue of M mod 6 12n.
A3.  For real a, b define the sequence x0, x1, x2, ... by x0 = a, xn+1 = xn + b sin xn. If b = 1, show that the sequence converges to a finite limit for all a. If b > 2, show that the sequence diverges for some a.
B1.  Find the maximum value of 1/√x + 2/√y + 3/√z, where x, y, z are positive reals satisfying 1/√2 ≤ z ≤ min(x√2, y√3), x + z√3 ≥ √6, y√3 + z√10 ≥= 2√5.
B2.  Find all real-valued continuous functions defined on the interval (-1, 1) such that (1 - x2) f(2x/(1 + x2) ) = (1 + x2)2 f(x) for all x.
B3.  a1, a2, ... , a2n is a permutation of 1, 2, ... , 2n such that |ai - ai+1| ≠ |aj - aj+1| for i ≠ j. Show that a1 = a2n + n iff 1 ≤ a2i ≤ n for i = 1, 2, ... n.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
15 June 2002