https://www.imo-official.org/problems/IMO2013SL.pdf
I'm wondering if there's a mistake in the first of the solutions given to A1, or rather that the proof is just insufficient. I agree that the given expression can be proved by induction and then I thought something interesting was about to happen, but instead it just says,
"applying (1) to the sequence given by $b_k=a_{n-k}$, the expressions coincide"
That seems to me to be more or less restating what we're trying to prove. Yes the reflected products are of the same form with a minimal index gap of 2 but there's no justification for why they should coincide.
Am I missing something?