39th Vietnamese Mathematical Olympiad 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 = a^N + b^N, 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, x_n+1 = x_n + b sin x_n. 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, ... , a_2n is a permutation of 1, 2, ... , 2n such that |a_i - a_i+1| ≠ |a_j - a_j+1| for i ≠ j. Show that a₁ = a_2n + n iff 1 ≤ a_2i ≤ n for i = 1, 2, ... n.

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