Here is the summary what you need to do to solve the problem. Introduce the wave amplitudes propagating in the fibres as follows (it will be convenient to use complex amplitudes, but vector diagrams for the amplitude additions will work, too). The main-frequency-waves:
E_B – the wave in the circular fibre propagating from A to C through B;
E_F – the wave in the circular fibre propagating from C to A through F, the diametrically opposite point to B;
E_D – the outgoing wave in the straight fibre towards D;
E_G – the outgoing wave in the upper straight fibre propagating leftwards from C. For all these waves we use the values at the beginning of their path, e.g. for E_B – the amplitude near the point A. With the help of the parameter \kappa introduced with the Hint 3, you should be able to obtain a closed set of equations for these wave amplitudes. The fact that there is an integer number of waves in the ring, that part of the wave E_F which continues propagating in the ring (contributing to E_B) adds up constructively with that part of the wave E_0 which jumps over into the ring. This means that, according to the Hint 1, that part of the wave E_F which jumps over to the straight fibre (contributing to E_D) adds up destructively with that part of the wave E_0 which continues propagating in the straight fibre. It is obvious that in order to maximize \mathcal I, you need to maximize the amount of wave power which is pumped into the energy of the double-frequency waves in the ring; this power can be expressed in terms of E_F, E_B, and \kappa. Alternatively, relying on the energy conservation law, you can minimize the total energy carried away by waves E_D and E_G. Note that E_F\approx E_B because the leaked-out part of these waves is small (\alpha \ll 1). This means, in particular, that the same value of \kappa can be used for the both semi-arcs. Now you should be able to find the optimal value of \kappa which maximizes the energy pumped into the double-frequency waves, together with the optimal values of E_F and E_B. With these values, it will be already relatively easy to find the requested quantities.
There will be no more hints, but you’ll have time until 13:00 GMT, 20th June to submit a solution for Problem 5. I recommend doing it before 23:59 GMT, 15th June as in that case you’ll get a feedback and will be having one more chance to correct your solution if it is not correct.
Problem 4 is now closed for submissions, the results and best solutions of Problem 4 will be published within a couple of days. The overall winners will be announced and the best solutions of Problem 5 will be published by 19th June (when the EuPhO-2021 will start).
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