42nd IMO 2001
A1. ABC is acute-angled. O is its circumcenter. X is the foot of the perpendicular from A to BC. Angle C = angle B + 30o. Prove that angle A + angle COX < 90o.
A2. a, b, c are positive reals. Let a’ = v(a2 + 8bc), b’ = v(b2 + 8ca), c’ = v(c2 + 8ab). Prove that a/a’ + b/b’ + c/c’ = 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.