Toán học, Đề thi toán vô địch thế giới, 2001

A1. ABC is acute-angled. O is its circumcenter. X is the foot of the perpendicular from A to BC. Angle C e" angle B + 30o. Prove that angle A + angle COX < 90o.

A2. a, b, c are positive reals. Let a' = �"(a2 + 8bc), b' = �"(b2 + 8ca), c' = �"(c2 + 8ab). Prove that a/a' + b/b' + c/c' e" 1.

A3. Integers are placed in each of the 441 cells of a 21 x 21 array. Each row and each column has at most 6 different integers in it. Prove that some integer is in at least 3 rows and at least 3 columns.

B1. Let n1, n2, ... , nm be integers where m is odd. Let x = (x1, ... , xm) denote a permutation of the integers 1, 2, ... , m. Let f(x) = x1n1 + x2n2 + ... + xmnm. Show that for some distinct permutations a, b the difference f(a) - f(b) is a multiple of m!.

B2. ABC is a triangle. X lies on BC and AX bisects angle A. Y lies on CA and BY bisects angle B. Angle A is 60o. AB + BX = AY + YB. Find all possible values for angle B.

B3. K > L > M > N are positive integers such that KM + LN = (K + L - M + N)(-K + L + M + N). Prove that KL + MN is composite.