This should not be on the D-list; it's an inequality of fairly reasonable interest to the inequality community and the OP has mentioned trying the obvious -- what other context would we expect?
Frankly, I think the general consensus is a lot less radical than you are on PSQs when they are of interest to more people than the OP.
But yes, on inequalities and triangle geometry it seems to differ the most.
My impression is, a lot of people come here from AoPS, where context is unusual (sometimes mods delete contexts in order to keep the OP "pure" for possible inclusion in contest lists).
It's a sudden and badly-documented change in expectations.
But even apart from that, math olympiad people find it annoying to protocol their failed attempts -- most of the time, there is no such thing as "a bit of progress".
I think a post with context is better than one without, unless the context is complete crap (my personal threshold for deleting context is when it is actively confusing, such as wrong definitions or nonsense) -- but that's not something commonly understood, methinks.
@darijgrinberg usually those problems come from somewhere though. At least the source should be specified and it should be placed explicitly in the context of competition mathematics.
Depending on context one or the other answer is presumably the desired one.
Now, some might say who cares lets give both or all, but that's not always desirable, as far as I am concerned.
Anyway, what seem clear is that context could change something for the question, contrary to claims that it's not useful.
Tangentially, the Lagrange multiplier answer also seems to show that the question is not really interesting at all except maybe as to show the value of learning some actual mathematics rather than spending ones time fiddling around with the same handful of tricks.
2
It's a shame that at high-school level often that type of pseudo-mathematics is the only thing that is offered to interested students.
@darijgrinberg to be clear regarding this at the risk of repetition: the source of the question, as incidentally requested in a comment years ago (by somebody that seems not generally hostile to this type of content).
@darijgrinberg same for this. :-)
@darijgrinberg this does include a bit of what I'd consider relevant context. Excluding derivatives as tools and indicating their background (10th grade).
"Use Lagrange multipliers" is a meaningless cheap comment. 50% of inequalities can be done with them (if you have a computer). It's like saying "have you tried induction?" or "use Cartesian coordinates". I often flag such comments for deletion if they don't come with any more pertinent content.
@darijgrinberg what's your point? The question is whether the method is admissible or not. In the current case the application seems straightforward. But that's irrelevant to the point. The problem is that due to lack of context it is not clear if the method is admissible in the given context. It might be. Or it might not be. Could be a contest problem, could also be calculus homework.
4
On the rest, well, so why do those problems?
Same for all those 'hard' integrals.
Seems just like a waste of time, in any case pseudo-mathematics.