2nd Putnam 1939 Problem A4 Given 4 lines in Euclidean 3-space: L 1 : x = 1, y = 0; L 2 : y = 1, z = 0; L 3 : x = 0, z = 1; L 4 : x = y, y = -6z. Find the equations of the two lines which both meet all of the L i .
A routine computation. Assume the line meets L 1 at (1,0,a) and L 2 at (b,1,0). Then it is (x - 1) = t(x - b), y = t(y - 1), (z - a) = tz. So it can only cut L 3 if 1/b = 1 - a, and L 4 if 6a = 6ab - 1. This gives a quadratic for a, which we can solve to get a = -1/2 or 1/3. Hence the possible lines are (1,0,-1/2) + t(-1/3,1,1/2) and (1,0,1/3) + t(1/2,1,-1/3). 2nd Putnam 1939 © John Scholes jscholes@kalva.demon.co.uk 4 Sep 1999