2. Closed convex curves $C_{1}^{\prime}, C_{2}^{\prime}$ have perimeters $x_{1}, x_{2}$, where $x_{1}+x_{2}=d$ $=$ const and are similar to curves $C_{1}, C_{2}$ which have perimeters $O_{1}, O_{2}$ and enclose areas with areas $P_{1}, P_{2}$, respectively. Determine $x_{1}, x_{2}$ such that the sum of the areas of the surfaces enclosed by the curves $C_{1}^{\prime}, C_{2}^{\prime}$ is minimal.
Solution. If $k_{1}$, respectively $k$ denote the similarity coefficient of the curves $C_{1}^{\prime}$ and $C_{1}$, respectively $C_{2}^{\prime}$ and $C_{2}$, then $x_{1}=k_{1} O_{1}$ and $x_{2}=k_{2} O_{2}$. Let $P_{1}^{\prime}, P_{2}^{\prime}$ be the areas of the regions bounded by the curves $C_{1}^{\prime}, C_{2}^{\prime}$, respectively. Then $P_{1}^{\prime}=k_{1}^{2} P_{1}, P_{2}^{\prime}=k_{2}^{2} P_{2}$, so the sum

$$
P_{1}^{\prime}+P_{2}^{\prime}=k_{1}^{2} P_{1}+k_{2}^{2} P_{2}=\frac{P_{1}}{O_{1}^{2}} x_{1}^{2}+\frac{P_{2}}{O_{2}^{2}} x_{2}^{2}=\left(\frac{P_{1}}{O_{1}^{2}}+\frac{P_{2}}{O_{2}^{2}}\right) x_{1}^{2}-\frac{2 d P_{2}}{O_{2}^{2}} x_{1}+\frac{d^{2} P_{2}}{O_{2}^{2}}
$$

is minimal for

$$
x_{1}=\frac{P_{2} O_{1}^{2}}{P_{2} O_{1}^{2}+P_{1} O_{2}^{2}} d, \quad x_{2}=\frac{P_{1} O_{2}^{2}}{P_{2} O_{1}^{2}+P_{1} O_{2}^{2}} d
$$