. Determine all values of the real number $a$ for which there exist five non-negative real numbers $x_{1}, \ldots, x_{5}$ satisfying the following relations:

$$
\sum_{k=1}^{5} k x_{k}=a, \sum_{k=1}^{5} k^{3} x_{k}=a^{2}, \sum_{k=1}^{5} k^{5} x_{k}=a^{3}
$$
. According to the Cauchy-Schwarz inequality, we have

$$
\left(\sum_{k=1}^{5} k^{3} x_{k}\right)^{2} \leqslant\left(\sum_{k=1}^{5} k x_{k}\right)\left(\sum_{k=1}^{5} k^{5} x_{k}\right)
$$

By hypothesis, we are in the case of equality. Therefore, for all $k$ such that $x_{k}$ is non-zero, the quotient $k^{5} x_{k} / k x_{k}$ must be a constant. At most one of the $x_{k}$ can thus be non-zero. Returning to the formulas in the statement, we find that for such a $k$ we have $x_{k}=k$, and a small case-by-case study then shows that the only possible values of $a$ are

$$
0,1,4,9,16,25
$$

Remark. We can also start from the following relation:

$$
a^{2} \sum_{k=1}^{5} k_{k}^{x}+\sum_{k=1}^{5} k^{5} x_{k}=2 a \sum_{k=1}^{5} k^{3} x_{k}
$$

which also gives

$$
\sum_{k=1}^{5} k\left(a-k^{2}\right)^{2} x_{k}=0
$$