## Task B-4.5.

For real numbers $x_{1}, x_{2}, \ldots, x_{2016}$, the following equalities hold:

$$
\frac{x_{1}}{x_{1}+1}=\frac{x_{2}}{x_{2}+2}=\cdots=\frac{x_{2016}}{x_{2016}+2016}, \quad x_{1}+x_{2}+x_{3}+\cdots+x_{2016}=2017
$$

Calculate $x_{1008}$.
## Solution.

Notice

$$
\frac{x_{1}}{x_{1}+1}=\frac{x_{i}}{x_{i}+i}, \text { for all } i \in\{1,2, \ldots, 2016\}
$$

Then $x_{1}(x_{i}+i)=x_{i}(x_{1}+1)$, so $x_{i}=i \cdot x_{1}$ for all $i \in\{1,2, \ldots, 2016\}$. It follows that

$$
\begin{array}{cc}
x_{1}+x_{2}+\cdots+x_{2016}=2017 & \\
x_{1} \cdot(1+2+\cdots+2016)=2017 & \\
x_{1} \cdot \frac{(1+2016) \cdot 2016}{2}=2017 & 1 \text{ point} \\
x_{1}=\frac{1}{1008} & 1 \text{ point} \\
x_{1008}=1008 \cdot x_{1}=1 . & 1 \text{ point}
\end{array}
$$