Question 34
Find all solutions in real numbers $x_{1}, x_{2}, \ldots, x_{n+1}$ all at least $1$ such that:
(1) $x_{1}^{1/2} + x_{2}^{1/3} + x_{3}^{1/4} + \ldots + x_{n}^{1/(n+1)} = n x_{n+1}^{1/2}$;
and
(2) $\dfrac{x_{1} + x_{2} + \ldots + x_{n}}{n} = x_{n+1}$.
(1) $x_{1}^{1/2} + x_{2}^{1/3} + x_{3}^{1/4} + \ldots + x_{n}^{1/(n+1)} = n x_{n+1}^{1/2}$;
and
(2) $\dfrac{x_{1} + x_{2} + \ldots + x_{n}}{n} = x_{n+1}$.