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**Bùi Thanh Tùng**

(tbhnams2002)

##### Thành viên danh dự

**41st Vietnam 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 < p. Define g(x) = f(x) f(1-x) for -1 <= x <= 1. Find the maximum and minimum values of f 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 <= |ak - k | <= 2 for all k. Show that ¾ 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) >= 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+.