11th Putnam 1951 Problem A2 k is a positive real and P 1 , P 2 , ... , P n are points in the plane. What is the locus of P such that ∑ PP i 2 = k? State in geometric terms the conditions on k for such points P to exist.
Let P i have coordinates (a i , b i ). Take the origin at the centroid of the points, so that ∑ a i = ∑ b i = 0. Then if P has coordinates (x, y), we have ∑ PP i 2 = ∑( (x - a i ) 2 + (y - b i ) 2 ) = n (x 2 + y 2 ) + ∑(a i 2 + b i 2 ). Let the origin be O, so that ∑(a i 2 + b i 2 ) = ∑ OP i 2 . Then if ∑ OP i 2 > k, the locus is empty. Otherwise it is the circle with centre at the centroid of the n points and radius √( (k - ∑ OP i 2 )/n ). 11th Putnam 1951 © John Scholes jscholes@kalva.demon.co.uk 5 Mar 2002