CMO 2016 Problem 2

Consider the following system of 10 equations in 10 real variables $v_{1}, \ldots, v_{10}$ :

$$v_{i}=1+\frac{6 v_{i}^{2}}{v_{1}^{2}+v_{2}^{2}+\cdots+v_{10}^{2}} (i=1, \ldots, 10) .$$

Find all 10-tuples $\left(v_{1}, v_{2}, \ldots, v_{10}\right)$ that are solutions of this system.