Question 84
Let $x_1, x_2, \dots, x_{100}$ be real numbers such that
$$
\begin{aligned}
|2x_k - x_{k+1}| &= x_{k+2} \quad \text{for all } k \in \{1, 2, \dots, 98\}, \\
|2x_{99} - x_{100}| &= x_1, \\
|2x_{100} - x_1| &= x_2.
\end{aligned}
$$
Prove that $x_1 = x_2 = \dots = x_{100}$.
(Tonći Kokan)
$$
\begin{aligned}
|2x_k - x_{k+1}| &= x_{k+2} \quad \text{for all } k \in \{1, 2, \dots, 98\}, \\
|2x_{99} - x_{100}| &= x_1, \\
|2x_{100} - x_1| &= x_2.
\end{aligned}
$$
Prove that $x_1 = x_2 = \dots = x_{100}$.
(Tonći Kokan)