Question 135
A positive integer $n$ is given. Determine all positive real numbers $x$ with
$$
n x^{2} + \frac{2^{2}}{x+1} + \frac{3^{2}}{x+2} + \ldots + \frac{(n+1)^{2}}{x+n} = n x + \frac{n(n+3)}{2}.
$$
$$
n x^{2} + \frac{2^{2}}{x+1} + \frac{3^{2}}{x+2} + \ldots + \frac{(n+1)^{2}}{x+n} = n x + \frac{n(n+3)}{2}.
$$