Question 113
Find all triples $(x, y, z)$ of positive real numbers $x$, $y$, $z$ for which the triples
$$
(23x + 24y + 25z, 23y + 24z + 25x, 23z + 24x + 25y)
$$
and
$$
(x^5 + y^5, y^5 + z^5, z^5 + x^5)
$$
are permutations of each other.
$$
(23x + 24y + 25z, 23y + 24z + 25x, 23z + 24x + 25y)
$$
and
$$
(x^5 + y^5, y^5 + z^5, z^5 + x^5)
$$
are permutations of each other.