Averages in a sequence III

William McWorter, Jr.
Mon, 23 January 2006

Let x₁, ..., x₁₀₀ be a sequence of real numbers. Suppose that for every subsequence of 8 terms, there exists a subsequence of 9 terms with the same average as that of the 8. Show that all x_i are equal.

Solution

First assume that the x_i are rational numbers. Multiplying each term by an appropriate fixed integer, we get a sequence of 100 integers still satisfying the average condition. This integer sequence is constant exactly when the rational sequence it came from is constant. So assume the x_i are integers.

Let S be the sum of any 8 of the x_i. Then there is a subsequence of 9 of the x_i, with sum T with the same average. Hence S/8 = T/9, or 9S = 8T. Thus S is divisible by 8. So every subsequence of 8 of the x_i has a sum divisible by 8.

Now let x_i and x_j be any two terms of the 100 terms and let M be the sum of any 7 terms other than x_i and x_j. Then M + x_i and M + x_j are sums of 8 terms and so are both divisible by 8. Hence x_i and x_j have the same remainder upon division by 8. Thus all 100 terms of the sequence have the same remainder modulo 8.

Suppose, by way of contradiction, that not all of the x_i are equal and let d > 0 be the smallest difference between two of the x_i among all integer sequences satisfying the average condition. Assume the x_i form a sequence with smallest difference between terms equal to d > 0. If we subtract the remainder modulo 8 from each term, we get another sequence satisfying the average condition with each term divisible by 8. Now divide each term of this new sequence by 8, obtaining a new integer sequence satisfying the average condition, but this time with smallest difference between terms equal to d/8 < d, contradicting our assumption that d > 0 is least among all sequences satisfying the average condition.

Thus, when all the x_i's are integers, the x_i's must be equal. Therefore, when all the x_i's are rational, the x_i's must be equal.

But now we can claim that all the x_i's are equal EVEN IF THE x_i's ARE REAL! For, the average condition defines a rational homogeneous linear system of equations which the x_i must satisfy. Our argument above shows that this linear system has rank 99 as a linear system of equations with rational coefficients. Hence the system has rank 99, even allowing real x_i's!

Evolution of a Solution

• Tue, 25 Mar 2003
• Sat, 16 July 2005
• Mon, 23 January 2006

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