The first time I tried to solve the problem, I thought I had a solution, but it turned out to be wrong. I wrongly assumed that would be maximized when , which is commonly true in Olympiad problems, but that needn’t be the case.
I then looked at solutions available online, and realized that I just needed to show that . After doing that, I homogenized both sides, and tried to prove the statement. I was finally successful. I am recording my solution, as it is slightly different from the ones available online.
Fruits of procrastination 