Example 7. Find all real numbers $a$ such that there exist non-negative real numbers $x_{\mathfrak{k}}, k=1,2,3,4,5$, satisfying the relations
$$
\begin{array}{c}
\sum_{\mathrm{k}=1}^{5} k x_{\mathrm{k}}=a, \quad \sum_{\mathrm{k}=1}^{5} k^{3} x_{k}=a^{2}, \\
\sum_{\mathrm{k}=1}^{5} k^{5} x_{\mathrm{k}}=a^{3} .
\end{array}
$$
$($ imO- $-1-6)$
If there exists a number $a$ such that the above three equations hold for the corresponding $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, then we have
$$
\left(\sum_{k=1}^{5} k^{3} x_{k}\right)^{2}=\left(\sum_{k=1}^{5} k x_{k}\right)\left(\sum_{k=1}^{5} k^{5} x_{k}\right) \text {. }
$$

On the other hand, by the Cauchy-Schwarz inequality, we also have
$$
\begin{array}{l}
\left(\sum_{k=1}^{5} k^{3} x_{k}\right)^{2}=\left(\sum_{k=1}^{5} k^{\frac{1}{2}} \sqrt{x_{k}} \cdot \frac{5}{k^{2}} \sqrt{x_{k}}\right)^{2} \\
\leqslant\left(\sum_{k=1}^{5} k x_{k}\right)\left(\sum_{k=1}^{5} k^{5} x_{k}\right) .
\end{array}
$$
Equation (1) precisely indicates that the inequality (2) must hold with equality, which is true if and only if
$$
k^{2} \sqrt{x_{k}} / \sqrt{x_{k}}=\lambda, k=1,2,3,4,5 . \text { (3) }
$$

If at least two of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are non-zero, then equation (3) cannot hold. Therefore, at most one of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ can be non-zero.

If $x_{1}=x_{2}=x_{3}=x_{4}=x_{5}=0$, then $a=0$; if $x_{r}=c \neq 0, x_{j}=0, j \neq r$, then we have
$$
r c=a, r^{3} c=a^{2}, r^{5} c=a^{3} \text {. }
$$

From this, we see: $a=r^{2}, r=c$. But $1 \leqslant r \leqslant 5$, so we get $a=1,4,9,16,25$. Therefore, all $a$ that satisfy the problem's requirements are $0,1,4,9,16$, and 25.