Hi Codeforces!

Not so long ago, quite interesting ideas of the problems in geometry came to mind, but no matter how much I think, it was impossible to approach. Therefore, I will write them here, in hope that someone will help or suggest a possibly interesting fact for their solution. It is possible that some of the tasks can be well-known, but...

Task 1. Given a circle with radius $$$r$$$ and an array of integers of length $$$n$$$. It is necessary to say whether there is such a polygon, the lengths of whose sides in the order of traversal are this array, and at the same time the circle of radius $$$r$$$ could be completely placed inside it.

Task 2.(Obviously reducible to the first problem via binary search, but perhaps there is an unrelated solution method) For a given array of integers of length $$$n$$$ find the maximum radius $$$r$$$ of a circle that there exists a polygon whose side lengths in traversal order are this array, and this circle could be completely placed inside it.

Task 3. Similar to task 2, but let us have the ability to change any one value in the array to an arbitrary (possibly non-integer) number so that the polygon remains non-degenerate. Again, it is required to maximize $$$r$$$.

Task 4. Not related to the previous ones. There is a set of $$$n \geq 3$$$ points on the plane. It is allowed to perform the following operation at most $$$k$$$ times: select two points from the set and rotate one relative to the other by an arbitrary angle. It is required to maximize the area of ​​the convex hull of the resulting set of points.

Task 5.Similar to task 4, but select not two points, but a segment (with ends at points from the set) and a point from the set (possibly coinciding with the end of the segment) and rotate the segment relative to the point by any angle. Again, it is necessary to maximize the area of ​​the convex hull of the resulting set of points.