IMO 1985 LL MON51

Origin: MON

Problem

Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers. From f1 one constructs a sequence fk of sequences as follows: if fk = (c1, c2, . . . , cn), then fk+1 = (ci1, ci2, ci3 + 1, ci4 + 1, . . . , cin + 1), where (ci1, ci2, . . . , cin) is a permutation of (c1, c2, . . . , cn). Give a necessary and sufficient condition for f1 under which it is possible for fk to be a constant sequence (b1, b2, . . . , bn), b1 = b2 = \cdot \cdot \cdot = bn, for some k.