The set of pairs of positive reals (x, y) such that xy = yx 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 = k1/(k-1), y = kk/(k-1). This gives a curve which cuts y = x at x = lim k1/(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
15 Feb 2002