12.107 For which natural numbers $n$, does there exist a convex $n$-gon, whose sides all have different lengths, and the sum of the distances from any point inside the polygon to its sides (or their extensions) is the same.
[Solution] It is easy to verify that the sum of the distances from any point inside a regular $n$-sided polygon to its sides or their extensions is constant.

When $n \geqslant 5$, by slightly translating each side of the regular $n$-sided polygon towards the center of the polygon, such that the distances of the translations are all different, the resulting $n$-sided polygon will have $n$ sides of unequal lengths. Thus, the resulting convex $n$-sided polygon meets the requirements of the problem.

When $n=3$, for any scalene triangle $ABC$, let $AD$ be the angle bisector of $\angle BAC$. Draw a line $EF \perp AD$ through any point on $AD$, intersecting $AB$ and $AC$ at $E$ and $F$ respectively. Then the sum of the distances from all points on segment $EF$ to $AE$ and $AF$ is constant. However, since $\triangle ABC$ is a scalene triangle, $EF$ is not parallel to $BC$, so the distances from different points on $EF$ to $BC$ are not equal. Therefore, the distances from different points on $EF$ to the three sides of $\triangle ABC$ are not equal. This shows that a triangle satisfying the requirements does not exist when $n=3$.

When $n=4$, a convex quadrilateral with four unequal sides must have at least one pair of opposite sides that, when extended, intersect. If the other pair of opposite sides are parallel, the quadrilateral is a trapezoid, which clearly cannot make the sum of the distances from all points inside to the four sides equal. Therefore, the other pair of opposite sides must also intersect. Draw the angle bisector $EG$ of the angle formed by the intersecting pair of opposite sides, and draw a line perpendicular to $EG$ through a point on $EG$, intersecting $AB$ and $CD$ at $K$ and $H$ respectively. Then the sum of the distances from all points on segment $HK$ to $AB$ and $CD$ is constant. However, $HK$ cannot be perpendicular to the angle bisector of the other intersecting angle, so the distances from points on $HK$ to $AD$ and $BC$ are not equal, and thus the distances from points on $HK$ to the four sides of the quadrilateral are not equal. This shows that a quadrilateral satisfying the requirements does not exist when $n=4$.

In summary, a polygon satisfying the requirements exists if and only if $n \geqslant 5$.