MathDB

51

Part of 1985 IMO Longlists

Problems(1)

Constant sequence (Bothered me a lot to type it)

Source:

9/13/2010
Let f1=(a1,a2,…,an),n>2f_1 = (a_1, a_2, \dots , a_n) , n > 2, be a sequence of integers. From f1f_1 one constructs a sequence fkf_k of sequences as follows: if fk=(c1,c2,…,cn)f_k = (c_1, c_2, \dots, cn), then fk+1=(ci1,ci2,ci3+1,ci4+1,...,cin+1)f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1), where (ci1,ci2,…,cin)(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}}) is a permutation of (c1,c2,…,cn)(c_1, c_2, \dots, c_n). Give a necessary and sufficient condition for f1f_1 under which it is possible for fkf_k to be a constant sequence (b1,b2,…,bn),b1=b2=⋯=bn(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n, for some k.k.
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