<s>[INST] $ \sum_{i=1}^{10} x_i =5 $, and $ \sum_{i=1}^{10} x_i^2 =6.1 $ ; Find the largest value of these ten numbers 
Ten non-negative numbers are such that  $ \sum_{i=1}^{10} x_i =5 $,
and $ \sum_{i=1}^{10} x_i^2 =6.1 $. What is the greatest value of the
largest of these numbers?

It presents a lot of confusion to me, but I think the largest number maybe a integer. Please help me with procedure.
Source : Moscow Institute of Physics and Technology admission assessment for computer Science
 [/INST] A General Problem
Suppose
$$
\frac{\sum_{i = 1}^n x_i}{n} = a
$$
and
$$
\frac{\sum_{i = 1}^n x_i^2}{n} = b^2
$$
Set $y_i = x_i - a$. Then (as you can check)
$$
\frac{\sum_{i = 1}^n y_i}{n} = 0
$$
So
$$
b^2 = \frac{\sum_{i = 1}^n (y_i + a)^2}{n} = \frac{\sum_{i = 1}^n  y_i^2 + 2ay_i + a^2}{n} = \frac{\sum_{i = 1}^n y_i^2}{n} + 2a \frac{\sum_{i = 1}^n y_i}{n} + \frac{a^2}{n} \sum_{i = 1}^n 1
$$
Setting
$$
c^2 = \frac{\sum_{i = 1}^n y_i^2}{n}
$$
We obtain
$$
b^2 = c^2 + 2a(0) + a^2
$$
So
$$
c^2 = b^2 - a^2
$$
Observe that such a sequence $y_i$ achieves its largest possible element precisely when $x_i$ achieves its largest possible element.
The upshot of all of this is that we've reduced the problem to finding the largest possible value of a sequence of numbers $y_i$ with a mean of $0$ (i.e. $\frac{\sum_{i =1}^n y_i}{n} = 0$) and variance of $c^2$ (i.e. $\frac{\sum_{i = 1}^n y_i^2}{n} = c^2$).
Intuitively, it's clear that the maximum should occur when $y_1 = \ldots = y_{n -1} = y$. Applying the mean and variance condition and performing a straightforward calculation yields $y_n = \sqrt{n - 1} c$. It remains to check that this is the best we can do.
Suppose for contradiction $y_n > \sqrt{n - 1} c$. Then
$$
\sum_{i = 1}^{n - 1} y_i = -y_n
$$
and
$$
\sum_{i = 1}^{n - 1} y_i^2 = nc^2 - y_n^2
$$
Thus
$$
\frac{\sum_{i =1}^{n - 1} y_i}{n -1} = -\frac{y_n}{n - 1}
$$
and
$$
\frac{\sum_{i = 1}^{n - 1} y_i^2}{n - 1} = \frac{nc^2 - y_n^2}{n - 1}
$$
Set $z_i = y_i + \frac{y_n}{n - 1}$. Then
$$
0 \leq \frac{\sum_{i = 1}^{n - 1} z_i^2}{n - 1} = \frac{nc^2 - y_n^2}{n - 1} - \left(\frac{y_n}{n - 1}\right)^2 < \frac{nc^2 - (n - 1)c^2}{n -1} - \frac{(n - 1)c^2}{(n - 1)^2} = 0
$$
which is absurd. Thus $y_n = \sqrt{n - 1} c$ is the best we can do.
In terms of $x_n$, $a$, and $b$ this is
$$
x_n = a + \sqrt{(n - 1)(b^2 - a^2)}
$$
Your Problem
In your problem $a = .5$ and $b^2 = .61$. Plugging in, you find that
$$
x_n = 2.3
$$
is the best you can do.
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