8. determine all integers $n$ for which there are positive real numbers $0<a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ exist with

$$
\sum_{k=1}^{n} a_{k}=96, \quad \sum_{k=1}^{n} a_{k}^{2}=144, \quad \sum_{k=1}^{n} a_{k}^{3}=216
$$

## Solution
For positive numbers $a_{k}$ the following applies according to CS

$$
\left(\sum_{k=1}^{n} a_{k}\right)\left(\sum_{k=1}^{n} a_{k}^{3}\right) \geq\left(\sum_{k=1}^{n} a_{k}^{2}\right)^{2}
$$

If you insert the given values for the three sums here, then equality applies. According to the general equality conditions for CS, there is therefore a positive number $\lambda$ with
$a_{k}^{3}=\lambda^{2} a_{k}$ for $1 \leq k \leq n$. This is equivalent to $a_{k}^{2}=\lambda^{2}$, i.e. to $a_{k}=\lambda$, since all $a_{k}$ are positive. If you insert $a_{1}=\ldots=a_{n}=\lambda$ into the three equations, then the following follows

$$
n \lambda=96, \quad n \lambda^{2}=144, \quad n \lambda^{3}=216
$$

The only solution is $\lambda=3 / 2$ and $n=64$.