IMO 1989 LL IND49

Origin: IND

Problem

Let A, B denote two distinct fixed points in space. Let X, P denote variable points (in space), while K, N, n denote positive integers. Call (X, K, N, P) admissible if (N −K)PA + K \cdot PB \geqN \cdot PX. Call (X, K, N) admissible if (X, K, N, P) is admissible for all choices of P. Call (X, N) admissible if (X, K, N) is admissible for some choice of K in the interval 0 < K < N. Finally, call X admissible if (X, N) is admissible for some choice of N (N > 1). Determine: (a) the set of admissible X; (b) the set of X for which (X, 1989) is admissible but not (X, n), n < 1989.