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 = 6^{n}, 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 12^{n}. | |
A3. For real a, b define the sequence x_{0}, x_{1}, x_{2}, ... by x_{0} = 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 - x^{2}) f(2x/(1 + x^{2}) ) = (1 + x^{2})^{2} f(x) for all x. | |
B3. a_{1}, a_{2}, ... , 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 a_{1} = a_{2n} + n iff 1 ≤ a_{2i} ≤ 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
jscholes@kalva.demon.co.uk
15 June 2002