mathisreal wrote: The sequence 0, 1, 1, 1, 1, 1,....,1 where have 1 number zero and 1995 numbers one. If we choose two or more numbers in this sequence(but not the all 1996 numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with 1996 numbers!!! Show that, we can repeat this operation until we have all 1996 numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!! Problem statement is not very clear : does the replaced number belong to the set of selected numbers or not ? 1) if it does, the problem is trivially wong : just consider the last step in the sequence of operations : it must start of a sequence where all numbers but one are identic and the replacement can never make these numbers all identic since it must be the replacement of the isolated number and this replacement can never transform it in the common number. 2) If it does not, the problem is trivially true : just take second and third numbers (1,1) and replace the first number (0) by the arithmetic mean of the selected numbers (1).

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