Riemann surface
matlab

How many sides does a convex polygon have if all its external angles are obtuse?

Solution to Problem 2.
$\left.\begin{array}{c}\text { Let } n=4 \ x_1, x_2, x_3, x_4 \& \text { ext }\end{array} \Rightarrow \begin{array}{c}x_1>90^{\circ} \ \vdots \ x_3>90^{\circ}\end{array}\right} \Rightarrow x_1+x_2+x_3+x_4>360^{\circ}$, so $n=4$ is impossible.
Therefore, $n=3$.

Prove that the ratio of the perimeters of two similar polygons is equal to their similarity ratio.

Solution to Problem 13.
$$L=P_1 P_2 \ldots, P_n ; L^{\prime}=P_1^{\prime} P_2^{\prime} \ldots, P_n^{\prime}$$
$L \sim L^{\prime} \Rightarrow(\exists) K>0$ and $f: L \rightarrow L^{\prime}$ such that $|P Q|=k|f(P) f(Q)|(\forall) P, Q \in L_{\text {, }}$ and $P_I^{\prime}=f\left(P_i\right)$.
Taking consecutively the peaks in the role of $P$ and $Q$, we obtain:
$$\left.\begin{array}{c} \left|P_1 P_2\right|=k\left|P_1^{\prime} P_2^{\prime}\right| \Rightarrow \frac{\left|P_1 P_2\right|}{\left|P_1^{\prime} P_2^{\prime}\right|}=k \ \left|P_2 P_3\right|=k\left|P_2^{\prime} P_3^{\prime}\right| \Rightarrow \frac{\left|P_2 P_3\right|}{\left|P_2^{\prime} P_3^{\prime}\right|}=k \ \vdots \ \left|P_{n-1} P_n\right|=k\left|P_{n-1}^{\prime} P_n^{\prime}\right| \Rightarrow \frac{\left|P_{n-1} P_n\right|}{\left|P_{n-1}^{\prime} P_n^{\prime}\right|}=k \ \left|P_n P_1\right|=k\left|P_n^{\prime} P_1^{\prime}\right| \Rightarrow \frac{\left|P_n P_1\right|}{\left|P_n^{\prime} P_1^{\prime}\right|}=k \end{array}\right} \Rightarrow$$

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