17th Brasil 1995


A1. ABCD is a quadrilateral with a circumcircle center O and an inscribed circle center I. The diagonals intersect at S. Show that if two of O, I, S coincide, then it must be a square.
A2. Find all real-valued functions on the positive integers such that f(x + 1019) = f(x) for all x, and f(xy) = f(x) f(y) for all xy.
A3. Let p(n) be the largest prime which divides n. Show that there are infinitely many positive integers n such that p(n) < p(n+1) < p(n+2).
B1. A regular tetrahedron has side L. What is the smallest x such that the tetrahedron can be passed through a loop of twine of length x?
B2. Show that the nth root of a rational (for n a positive integer) cannot be a root of the polynomial x⁵ - x⁴ - 4x³ + 4x² + 2.
B3. X has n elements. F is a family of subsets of X each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of X with at least √(2n) members which does not contain any members of F.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.