k is a positive real and P1, P2, ... , Pn are points in the plane. What is the locus of P such that ∑ PPi2 = k? State in geometric terms the conditions on k for such points P to exist.
Let Pi have coordinates (ai, bi). Take the origin at the centroid of the points, so that ∑ ai = ∑ bi = 0. Then if P has coordinates (x, y), we have ∑ PPi2 = ∑( (x - ai)2 + (y - bi)2 ) = n (x2 + y2) + ∑(ai2 + bi2). Let the origin be O, so that ∑(ai2 + bi2) = ∑ OPi2. Then if ∑ OPi2 > k, the locus is empty. Otherwise it is the circle with centre at the centroid of the n points and radius √( (k - ∑ OPi2)/n ).
11th Putnam 1951
© John Scholes
5 Mar 2002