Points on a Sphere: 10855

Points on a Sphere 10855 [2001, 171]. Proposed by Ernesto Bruno Cossi, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil. Given n + 2 points P1, P2, ..., Pn+2 in R", let f(j) be the volume of the n-dimensional simplex whose vertex set is {P1, P2, ..., Pn+2} {Pj), and let g(j) = +f(j) according to whether the vertices P1, P2,... , Pj-1, Pj+, ... Pn+2 give the positive orientation on the simplex. Prove that the Pj lie on a common sphere in R" if and only if Ej+2(-l)ys(Pj)g(j) = 0, where s(P) is the square of the distance from the origin to the point P. Solution by Robin Chapman, University of Exeter, Exeter, U. K. For the problem statement to be correct, we must replace the phrase "common sphere" with "common sphere or hyperplane." Denote the coordinates of a point Q in IRn by (x (Q), ... , X (Q)). If Q1, ... , Qn+ are n + 1 points in Rn, then the determinant 1 xl(QI) ... xn(Qi) 1 xl(Q2) ... Xn(Q2) D(Ql,..., Qn+l) =