1. Solve the system $\left\{\begin{array}{l}\sqrt{2 x^{2}+2}=y+1 ; \\ \sqrt{2 y^{2}+2}=z+1 ; \\ \sqrt{2 z^{2}+2}=x+1 .\end{array}\right.$
Answer: $(1,1,1)$.

Sketch of the solution. Squaring each of the equations and adding them, we move all the terms to the left side and factor out complete squares. We get $(x-1)^{2}+(y-1)^{2}+(z-1)^{2}=0$. From this, $x=y=z=1$.

Verification shows that this triplet works, as the left and right sides are both equal to 2.

Criteria. It is obtained that the triplet $(1,1,1)$ is a solution. However, no verification has been done: 5 points.

Criteria: Answer without justification: 1 point.