2nd Mexican 1988


A1. In how many ways can we arrange 7 white balls and 5 black balls in a line so that there is at least one white ball between any two black balls?
A2. If m and n are positive integers, show that 19 divides 11m + 2n iff it divides 18m + 5n.
A3. Two circles of different radius R and r touch externally. The three common tangents form a triangle. Find the area of the triangle in terms of R and r.
A4. How many ways can we find 8 integers a₁, a₂, ... , a₈ such that 1 ≤ a₁ ≤ a₂ ≤ ... ≤ a₈ ≤ 8?
B1. a and b are relatively prime positive integers, and n is an integer. Show that the greatest common divisor of a²+b²-nab and a+b must divide n+2.
B2. B and C are fixed points on a circle. A is a variable point on the circle. Find the locus of the incenter of ABC as A varies.
B3. [unclear]
B4. Calculate the volume of an octahedron which has an inscribed sphere of radius 1.

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