$ \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

Just apply Cauchy-Schwarz or AM-QM inequality to the nine smaller numbers, you get
(WLOG assume $x_1 \le x_2 \le \cdots \le x_{10}$)
$$
\left(\sum_{i=1}^9 x_i\right)^2 \le 9 \left( \sum_{i=1}^9 x_i^2\right)\\
\iff
(5-x_{10})^2 \le 9(6.1-x_{10}^2) \\
\iff 10x_{10}^2 - 10x_{10} - 29.9 \le 0 \\
\implies x_{10} \le 2.3.\blacksquare$$
