2. Find for all natural $n>1$ the positive solutions of the system

$$
\left\{\begin{array}{l}
x_{1}+2 x_{2}+\cdots+n x_{n}=3 \\
\frac{1}{x_{1}}+\frac{1}{2 x_{2}}+\cdots+\frac{1}{n x_{n}}=3
\end{array}\right.
$$
Answer. For $n=3: x_{1}=1, x_{2}=1 / 2, x_{3}=1 / 3$; for $\quad n=2: x_{1}=\frac{3 \pm \sqrt{5}}{2}, x_{2}=\frac{3 \mp \sqrt{5}}{4} . \quad$ For other values of $n$, no solutions exist.

Solution. Let $y_{k}=k x_{k}$ and add the equations of the system:

$$
\left(y_{1}+\frac{1}{y_{1}}\right)+\left(y_{2}+\frac{1}{y_{2}}\right)+\cdots+\left(y_{n}+\frac{1}{y_{n}}\right)=6
$$

For positive numbers, the inequality $a+\frac{1}{a} \geq 2$ holds (since $a^{2}+1-2 a \geq 0$). Therefore, the left side is not less than $2 n$, from which it follows that $n \leq 3$. For $n=3$, each term equals 2, hence $y_{1}=y_{2}=y_{3}=1$, and $x_{1}=1, x_{2}=1 / 2, x_{3}=1 / 3$. For $n=2$, the system becomes

$$
\left\{\begin{array}{l}
x_{1}+2 x_{2}=3 \\
\frac{1}{x_{1}}+\frac{1}{2 x_{2}}=3
\end{array}\right.
$$

$$
2 x_{2}=3-x_{1}, \frac{1}{x_{1}}+\frac{1}{3-x_{1}}=3, x_{1}=\frac{3 \pm \sqrt{5}}{2}, x_{2}=\frac{3 \mp \sqrt{5}}{4} \text {. }
$$

Comment. A correct solution earns 20 points, distributed as follows: 7 points for the solution for each found value of $n$, 6 points for proving that no solutions exist for $n>3$. A correct solution with one computational error earns 19 points. For $n=2$, if one pair of roots is lost, 1 point is deducted. Some progress in the solution earns 5 points. If the solution is started but the progress is insignificant, 1 point is awarded. An incorrect or missing solution earns 0 points.