What are the possible values of the sum of squares, given a specific sum?

[Ackbach]

Here is this week's POTW:

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Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\displaystyle\sum_{j=0}^\infty x_j=A$?

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[Ackbach]

Re: Problem Of The Week # 259 - Apr 17, 2017

This was Problem A-1 in the 2000 William Lowell Putnam Mathematical Competition.

No one answered this week's POTW. The solution, attributed to Kiran Kedlaya and his associates, follows:

Spoiler

The possible values comprise the interval $(0, A^2)$.

To see that the values must lie in this interval, note that
\[
\left(\sum_{j=0}^m x_j\right)^2
= \sum_{j=0}^m x_j^2 + \sum_{0\leq j<k\leq m} 2x_jx_k,
\]
so $\sum_{j=0}^m x_j^2 \leq A^2 - 2x_0x_1$. Letting $m \to \infty$, we have $\sum_{j=0}^\infty x_j^2 \leq A^2-2x_0x_1 < A^2$.

To show that all values in $(0, A^2)$ can be obtained, we use geometric progressions with $x_1/x_0 = x_2/x_1 = \cdots = d$ for variable $d$. Then $\sum_{j=0}^\infty x_j = x_0/(1-d)$ and
\[
\sum_{j=0}^\infty x_j^2 = \frac{x_0^2}{1-d^2} = \frac{1-d}{1+d} \left(
\sum_{j=0}^\infty x_j \right)^2.
\]
As $d$ increases from 0 to 1, $(1-d)/(1+d)$ decreases from 1 to 0. Thus if we take geometric progressions with $\sum_{j=0}^\infty x_j = A$, $\sum_{j=0}^\infty x_j^2$ ranges from 0 to $A^2$. Thus the possible values are indeed those in the interval $(0, A^2)$, as claimed.