5.1. Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2017} \underbrace{22 \ldots .22}_{2018} 5}$.
Answer: 6056 (the number from the condition of the problem is $\underbrace{33 \ldots 33}_{2017} 5$).

Solution. Since

$\underbrace{11 \ldots 11}_{2017} \underbrace{22 \ldots 22}_{2018} 5=\frac{10^{2017}-1}{9} \cdot 10^{2019}+\frac{10^{2018}-1}{9} \cdot 20+5=\frac{10^{4036}+10^{2019}+25}{9}=\left(\frac{10^{2018}+5}{3}\right)^{2}$,

the number from the condition of the problem is $\frac{10^{2018}+5}{3}=\overbrace{\frac{99 \ldots 99}{2018}+6}^{3}=\underbrace{33 \ldots 33}_{2018}+2=\underbrace{33 \ldots 3}_{2017} 5$, it is an integer, and the sum of its digits is $2017 \cdot 3+5=6056$.