5th Mexican 1991


A1. Find the sum of all positive irreducible fractions less than 1 whose denominator is 1991.
A2. n is palindromic (so it reads the same backwards as forwards, eg 15651) and n = 2 mod 3, n = 3 mod 4, n = 0 mod 5. Find the smallest such positive integer. Show that there are infinitely many such positive integers.
A3. 4 spheres of radius 1 are placed so that each touches the other three. What is the radius of the smallest sphere that contains all 4 spheres?
B1. ABCD is a convex quadrilateral with AC perpendicular to BD. M, N, R, S are the midpoints of AB, BC, CD, DA. The feet of the perpendiculars from M, N, R, S to CD, DA, AB, BC are W, X, Y, Z. Show that M, N, R, S, W, X, Y, Z lie on the same circle.
B2. The sum of the squares of two consecutive positive integers can be a square, for example 3² + 4² = 5². Show that the sum of the squares of 3 or 6 consecutive positive integers cannot be a square. Give an example of the sum of the squares of 11 consecutive positive integers which is a square.
B3. Let T be a set of triangles whose vertices are all vertices of an n-gon. Any two triangles in T have either 0 or 2 common vertices. Show that T has at most n members.

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