Notice that by using the Hints 1-3, this problem about the fastest lap can be reduced to another problem about the shape of a flexible inextensible rope the endpoints of which can slide frictionlessly along two rigid rails drawing straight lines y=\pm kx, with a suitably chosen tangent k.
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 2 will be published at 13:00 GMT, 30th January 2022, together with the next updates of the intermediate results.