353. Find all solutions of the system of equations

$$
\left\{\begin{array}{l}
y^{2}=x^{3}-3 x^{2}+2 x \\
x^{2}=y^{3}-3 y^{2}+2 y
\end{array}\right.
$$
353. Answer: $(0 ; 0),(2+\sqrt{2} ; 2+\sqrt{2}),(2-\sqrt{2} ; 2-\sqrt{2})$. Subtracting the second equation from the first, we get $f(x)=f(y)$, where the function $f(x)=x^{3}-2 x^{2}+2 x$ is increasing ( $f^{\prime}(x)>0$ ).

$$
\begin{aligned}
& \text { 354. If } k \geqslant a \underbrace{44 \ldots 45}_{n-2}, \quad(a \geqslant 1-\text { first digit }), \\
& \tilde{k}=(a+1) \cdot 10^{n-1} \text { and } \tilde{k} / k \leqslant \frac{(a+1) 10^{n-1}}{\underbrace{44}_{\underbrace{44}_{n-2} \ldots 4}}<\frac{a+1}{a+\frac{4}{9}} \leqslant \frac{18}{13} .
\end{aligned}
$$

If $k \leqslant \underbrace{a 44 \ldots 4}_{n-1}$, then $\tilde{k}=a 10^{n-1}$ and $\tilde{k} / k \leqslant 1$.

$$
\text { 355. } \begin{aligned}
S_{D E F} & =\frac{1}{2}\left(S_{B E F}+S_{A E F}\right)<\frac{1}{2}\left(S_{B E F}+S_{A D F}\right)= \\
& =\frac{1}{2} S_{A B F E}=\frac{1}{2}\left(S_{A B F}+S_{A F E}\right) \leqslant \frac{1}{2}\left(S_{A B F}+S_{A B E}\right)= \\
& =S_{B D F}+S_{A D E} .
\end{aligned}
$$