Problem 2. Option I.

Solve the system of equations $\left\{\begin{array}{l}p^{2}+q^{2}+r^{2}=6, \\ p q-s^{2}-t^{2}=3 .\end{array}\right.$
Solution. $\left\{\begin{array}{l}p^{2}+q^{2}+r^{2}=6, \\ 2 p q-2 s^{2}-2 t^{2}=6\end{array} \Rightarrow(p-q)^{2}+2 s^{2}+2 t^{2}+r^{2}=0\right.$.

From this $\left\{\begin{array}{l}p=q, \\ s=t=p=0 .\end{array}\right.$

From the equation $p^{2}+q^{2}=6$ we find $p=q= \pm \sqrt{3}$.

Answer: $(p ; q ; r ; s ; t) \in\{( \pm \sqrt{3} ; \pm \sqrt{3} ; 0 ; 0 ; 0)\}$.