A randomized algorithm for finding a hyperplane separating two finite point sets in the Euclidean space Rd and a randomized algorithm for solving linearly constrained general convex quadratic problems are proposed. The expected running time of the separating algorithm is O(dd! (m + n)), where m and n are cardinalities of sets to be separated. The expected running time of the algorithm for solving quadratic problems is O(dd! s) where s is the number of inequality constraints. These algorithms are based on the ideas of Seidel’s linear programming algorithm [6]. They are closely related to algorithms of [8], [2], and [9] and belong to an abstract class of algorithms investigated in [1]. The algorithm for solving quadratic problems has some features of the one proposed in [7].