3. Solve the system of equations:

$$
\left\{\begin{array}{l}
2 x^{2}+3 y+5=2 \sqrt{2 z+5} \\
2 y^{2}+3 z+5=2 \sqrt{2 x+5} \\
2 z^{2}+3 x+5=2 \sqrt{2 y+5}
\end{array}\right.
$$
Answer: $x=y=z=-0.5$.

Solution. Consider the function $f(t)=2 t^{2}+3 t+5-2 \sqrt{2 t+5}$. It can be rewritten as $f(t)=\frac{1}{2}(2 t+1)^{2}+\frac{1}{2}(\sqrt{2 t+5}-2)^{2}$. Clearly, $f(t) \geqslant 0$ and equality is achieved only at $t=-0.5$. If we add the left and right sides of our system, we get the equation $f(x)+f(y)+f(z)=0$, which has a solution only when $x=y=z=-0.5$.