Question 13
(ROM) Let $A_0,A_1,\ldots,A_n$ be points in a plane such that
(i) $A_0A_1\le\frac12A_1A_2\le\cdots\le\frac1{2^{n-1}}A_{n-1}A_n$ and
(ii) $0\lt \angle A_0A_1A_2\lt \angle A_1A_2A_3\lt \cdots\lt \angle A_{n-2}A_{n-1}A_n\lt 180^\circ$,
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0\le k\le m-2\lt n-2$.
(i) $A_0A_1\le\frac12A_1A_2\le\cdots\le\frac1{2^{n-1}}A_{n-1}A_n$ and
(ii) $0\lt \angle A_0A_1A_2\lt \angle A_1A_2A_3\lt \cdots\lt \angle A_{n-2}A_{n-1}A_n\lt 180^\circ$,
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0\le k\le m-2\lt n-2$.