41st Vietnamese Mathematical Olympiad 2003 Problems

41st Vietnamese Mathematical Olympiad 2003 Problems

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(1-x) for -1 ≤ x ≤ 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 S_n be the number of permutations (a₁, a₂, ... , a_n) of (1, 2, ... , n) such that 1 ≤ |a_k - k | ≤ 2 for all k. Show that (7/4) S_n-1 < S_n < 2 S_n-1 for n > 6.

B1.  Find the largest positive integer n such that the following equations have integer solutions in x, y₁, y₂, ... , y_n:

(x + 1)² + y₁² = (x + 2)² + y₂² = ... = (x + n)² + y_n².

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) ≥ 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⁺.

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