22nd Putnam 1961 Problem A1 The set of pairs of positive reals (x, y) such that x y = y x form the straight line y = x and a curve. Find the point at which the curve cuts the line.
Put y = kx. Then we get x = k 1/(k-1) , y = k k/(k-1) . This gives a curve which cuts y = x at x = lim k 1/(k-1) , where the limit is taken as k tends to 1. Put k = 1 + 1/n, then the limit is (1 + 1/n) n , which is e. So the point of intersection is (e, e). 22nd Putnam 1961 © John Scholes jscholes@kalva.demon.co.uk 15 Feb 2002