Croatian Mathematical Competitions

Squaring the given equations we get
$$\begin{gathered} 4x_k^2 - 4x_kx_{k+1} + x_{k+1}^2 = x_{k+2}^2 \quad \text{for } k \in \{1, 2, \dots, 98\}, \\ 4x_{99}^2 - 4x_{99}x_{100} + x_{100}^2 = x_1^2, \quad 4x_{100}^2 - 4x_{100}x_1 + x_1^2 = x_2^2. \end{gathered}$$
Adding these equations we get
$$4(x_1^2 + x_2^2 + \dots + x_{100}^2) - 4(x_1x_2 + \dots + x_{99}x_{100} + x_{100}x_1) = 0.$$
After regrouping and dividing by 2 we get
$$(x_1 - x_2)^2 + \dots + (x_{99} - x_{100})^2 + (x_{100} - x_1)^2 = 0.$$
Since all the terms are non-negative, they all have to be equal to zero, so we have $x_1 = \dots = x_{100}$.