### 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.

**Solution**

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