Question 197
Find all triples $(x, y, z)$ of positive real numbers such that
$$
\begin{aligned}
x^2 + 2y^2 &= x + 2y + 3z, \\
y^2 + 2z^2 &= 2x + 3y + 4z, \\
z^2 + 2x^2 &= 3x + 4y + 5z.
\end{aligned}
$$
(Patrik Bak)
$$
\begin{aligned}
x^2 + 2y^2 &= x + 2y + 3z, \\
y^2 + 2z^2 &= 2x + 3y + 4z, \\
z^2 + 2x^2 &= 3x + 4y + 5z.
\end{aligned}
$$
(Patrik Bak)