11th Putnam 1951

k is a positive real and P₁, P₂, ... , P_n are points in the plane. What is the locus of P such that ∑ PP_i² = k? State in geometric terms the conditions on k for such points P to exist.

Solution

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_i² = ∑( (x - a_i)² + (y - b_i)² ) = n (x² + y²) + ∑(a_i² + b_i²). Let the origin be O, so that ∑(a_i² + b_i²) = ∑ OP_i². Then if ∑ OP_i² > k, the locus is empty. Otherwise it is the circle with centre at the centroid of the n points and radius √( (k - ∑ OP_i²)/n ).

11th Putnam 1951

© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002