1. Compute the sum:

$$
\sqrt{2018-\sqrt{2019 \cdot 2017}}+\sqrt{2016-\sqrt{2017 \cdot 2015}}+\cdots+\sqrt{2-\sqrt{3 \cdot 1}}
$$
Answer: $\frac{\sqrt{2019}-1}{\sqrt{2}}$.

Solution: Using the known formula $\sqrt{a-\sqrt{(a+1)(a-1)}}=\frac{\sqrt{a+1}-\sqrt{a-1}}{\sqrt{2}}$, valid for $a \geq 1$, we get that each of the roots can be represented as:

$$
\begin{gathered}
\sqrt{2018-\sqrt{2019 \cdot 2017}}=\frac{\sqrt{2019}-\sqrt{2017}}{\sqrt{2}} \\
\sqrt{2016-\sqrt{2017 \cdot 2015}}=\frac{\sqrt{2017}-\sqrt{2015}}{\sqrt{2}} \\
\cdots \\
\sqrt{2-\sqrt{3 \cdot 1}}=\frac{\sqrt{3}-\sqrt{1}}{\sqrt{2}}
\end{gathered}
$$

Adding all these equalities, we have

$$
\sqrt{2018-\sqrt{2019 \cdot 2017}}+\sqrt{2016-\sqrt{2017 \cdot 2015}}+\cdots+\sqrt{2-\sqrt{3 \cdot 1}}=\frac{\sqrt{2019}-\sqrt{1}}{\sqrt{2}}
$$