ISL 1989 Problem 47

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 $$(N - K) \cdot PA + K \cdot PB \geq N \cdot PX.$$ 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 $(X, n), n < 1989.$