Regarding the assumption (b) of the problem statement: the heat flux along the wire cannot be neglected at those points which separate those regions of the wire which are in the high-resistivity state from those regions which are in the low-resistivity state. (Elsewhere, the heat flux along the wire can be neglected.)
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 1 will be published at 13:00 GMT, 8th January 2023, together with the next updates of the intermediate results.