36th Spanish 2000


A1. Let p(x) = x⁴ + ax³ + bx² + cx + 1, q(x) = x⁴ + cx³ + bx² + ax + 1. Find conditions on a, b, c (assuming a ≠ c) so that p(x) and q(x) have two common roots. In this case solve p(x) = q(x) = 0.
A2. The diagram shows a network of roads. The distance from one node to an adjacent node is 1. P goes from A to B by a path length 7, and Q goes from B to A by a path length 7. Each goes at the same constant speed. At each junction with two possible directions to take, each has probability 1/2. Find the probability that P and Q meet.
A3. Circles C and C' meet at A and B. P, P' are variable points on C, C' such that P, B, P' are collinear. Show that the perpendicular bisector of PP' passes through a fixed point M (which depends only on C and C').
B1. Find the largest integer N such that [N/3] has three digits, all equal, and [N/3] = 1 + 2 + 3 + ... + n for some n.
B2. Given 4 points inside or on the perimeter of a square side 1, show that two of them must be distance ≤ 1 apart.
B3. Let N be the set of positive integers. Show that there is no function f : N → N such that f(f(n)) = n+1 for all n.

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