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 xi are equal.

Solution

First assume that the xi 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 xi are integers.

Let S be the sum of any 8 of the xi. Then there is a subsequence of 9 of the xi, 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 xi has a sum divisible by 8.

Now let xi and xj be any two terms of the 100 terms and let M be the sum of any 7 terms other than xi and xj. Then M + xi and M + xj are sums of 8 terms and so are both divisible by 8. Hence xi and xj 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 xi are equal and let d > 0 be the smallest difference between two of the xi among all integer sequences satisfying the average condition. Assume the xi 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 xi's are integers, the xi's must be equal. Therefore, when all the xi's are rational, the xi's must be equal.

But now we can claim that all the xi's are equal EVEN IF THE xi's ARE REAL! For, the average condition defines a rational homogeneous linear system of equations which the xi 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 xi's!

Evolution of a Solution

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