51st Putnam 1990 Problem A4 Given a point P in the plane, let S P be the set of points whose distance from P is irrational. What is the smallest number of such sets whose union is the entire plane?
We clearly need at least 3 sets, since given any two points P and Q we can find a point R whose distance from each of P and Q is rational (indeed integral). However, 3 suffice. Take any two distinct points P and Q. There are at most two points with given rational distances x from P and y from Q, so there are only countably many points X outside S P and S Q . Each of the these points X rules out as the third point countably many points on the line PQ (namely those which are a rational distance from X). So only countably many points are ruled out in total from a line with uncountably many points. Take one of the remaining points. 51st Putnam 1990 © John Scholes jscholes@kalva.demon.co.uk 1 Jan 2001