A1. Let R be the reals and f: R → R a function such that f( cot x ) = cos 2x + sin 2x for all 0 < x < π. Define g(x) = f(x) f(1x) for 1 ≤ x ≤ 1. Find the maximum and minimum values of g on the closed interval [1, 1].  
A2. The circles C_{1} and C_{2} touch externally at M and the radius of C_{2} is larger than that of C_{1}. A is any point on C_{2} which does not lie on the line joining the centers of the circles. B and C are points on C_{1} such that AB and AC are tangent to C_{1}. The lines BM, CM intersect C_{2} again at E, F respectively. D is the intersection of the tangent at A and the line EF. Show that the locus of D as A varies is a straight line.


A3. Let S_{n} be the number of permutations (a_{1}, a_{2}, ... , a_{n}) of (1, 2, ... , n) such that 1 ≤ a_{k}  k  ≤ 2 for all k. Show that (7/4) S_{n1} < S_{n} < 2 S_{n1} for n > 6.  
B1. Find the largest positive integer n such that the following equations have integer solutions in x, y_{1}, y_{2}, ... , y_{n}:
(x + 1)^{2} + y_{1}^{2} = (x + 2)^{2} + y_{2}^{2} = ... = (x + n)^{2} + y_{n}^{2}. 

B2. Define p(x) = 4x^{3}  2x^{2}  15x + 9, q(x) = 12x^{3} + 6x^{2}  7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that A^{2} + 3 B^{2} = 4.  
B3. Let R^{+} be the set of positive reals and let F be the set of all functions f : R^{+} → R^{+} such that f(3x) ≥ f( f(2x) ) + x for all x. Find the largest A such that f(x) ≥ A x for all f in F and all x in R^{+}. 
Many thanks to Hung Ha Duy for providing and translating the questions.
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© John Scholes
jscholes@kalva.demon.co.uk
25 Mar 2003
Last updated/corrected 9 July 03