Toán học, Olympic toán toàn quốc - Việt nam 2003

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 < À�. Define g(x) = f(x) f(1-x) for -1 d" x d" 1. Find the maximum and minimum values of g on the closed interval [-1, 1].

A2. The circles C1 and C2 touch externally at M and the radius of C2 is larger than that of C1. A is any point on C2 which does not lie on the line joining the centers of the circles. B and C are points on C1 such that AB and AC are tangent to C1. The lines BM, CM intersect C2 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 Sn be the number of permutations (a1, a2, ... , an) of (1, 2, ... , n) such that 1 d" |ak - k | d" 2 for all k. Show that (7/4) Sn-1 < Sn < 2 Sn-1 for n > 6.

B1. Find the largest positive integer n such that the following equations have integer solutions in x, y1, y2, ... , yn: (x + 1)2 + y12 = (x + 2)2 + y22 = ... = (x + n)2 + yn2.

B2. Define p(x) = 4x3 - 2x2 - 15x + 9, q(x) = 12x3 + 6x2 - 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 A2 + 3 B2 = 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) e" f( f(2x) ) + x for all x. Find the largest A such that f(x) e" A x for all f in F and all x in R+.