This problem required skills of using small parameters dealing with seemingly divergent integrals. At the very end, all the accepted solutions followed the rout sketched by the hints. However, originally, there was one solution  provided by Nozomi Sakura  which was exact, but went mathematically well beyond the IPhO Syllabus, and was therefore rejected. Another reason for the rejection was that the exact answer was expressed in terms of modified Bessel of the second kind; finding its asymptotic behaviour for large permeability was not a trivial task and had been executed incorrectly. The solution was based on textbook exercises, (W.R. Symthe, Static and Dynamic Electricity). I feel that finding this textbook with hints for solving the problem exactly deserves a credit, so he receives 0.4 of the bonus. I am not linking here his exact solution, because aforementioned reasons (anyone interested in the exact solution is encouraged to read the Symthe's textbook). Instead, here is asymptotic solution for large permeability. Keep in mind, though, that this solution has mostly formulas and explanations are minimal; as a matter of fact, the other solutions linked below are explained much more carefully.
The rest of the bonus is distributed between those solutions which were submitted within first five weeks: Thomas Bergamaschi; Navneel Singhal; Dolteanu Stefan; Yunus Emre Parmaksiz. And here are the results. Number of fully correct solutions: 9.

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