I am announcing cryptocurrency bounties for solving some of my favorite problems in mathematics. Here is the first one:
- Let P=(v1, v2, v3, v4) be a simple polygon drawn on a plane.
- The co-ordinates of the vertices v1, v2, v3, v4 are all rational numbers.
- The lengths of the edges (v1, v2), (v2, v3), (v3, v4) and (v4, v1) are all integers.
- The distance between v1 and v3 is an integer.
Conjecture 1: There exists a point x with rational coordinates inside P such that the euclidean distances between the pairs (x, v1), (x, v2), (x, v3), (x, v4) are all rational numbers.
Conjecture 2: Same as Conjecture 1 when the polygon P is convex.
- 100 Eulercoins for proving Conjecture 1. This implies Conjecture 2 is also true.
- 50 Eulercoins for disproving only Conjecture 1.
- 50 Eulercoins for proving only Conjecture 2.
- 100 Eulercoins for disproving Conjecture 2. This implies Conjecture 1 is also false.
- The bounties are valid till Dec 31 2021. If they are not resolved by Dec 31 2021, I will revisit this and update the bounties.
If you have any questions (or) solutions (or) counter-examples, leave a comment.