IMO 1989 LL IRE56

Origin: IRE

Problem

Let n = 2k −1, where k \geq6 is an integer. Let T be the set of all n-tuples (x1, x2, . . . , xn) where xi is 0 or 1 (i = 1, 2, . . ., n). For x = (x1, . . . , xn) and y = (y1, . . . , yn) in T , let d(x, y) denote the number of integers j with 1 \leqj \leqn such that xj ̸= yj. (In particular d(x, x) = 0.) Suppose that there exists a subset S of T with 2k elements that has the following property: Given any element x in T , there is a unique element y in S with d(x, y) \leq3. Prove that n = 23.