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Olympiad inequality. Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the minimal value $P$ of $$f:=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$$ Note: Often Stack Exchange asks to show some work before answering the question. This inequality was used as a proposed problem for the National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They didn't believe that any student can solve this problem in the $3$ hour timeframe.

It seems that minimum is achieved at $(a,b,c)=(0,1,1).$ I've tried to prove $$f\ge \frac{\sqrt{5}}{4}+\frac{2\sqrt{2}}{5}. \tag{1}$$ A big problem around here is $a=b=c=\dfrac{\sqrt{3}}{3}$ since $LHS_{(1)}-RHS_{(1)}\approx 0.000151$

I posted it here. There is a proof by RiverLi but it's very complicated and not appropriate in contest time.

I hope to see some ideas. Thank you!

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    $\begingroup$ Could you write $f(a,b,c)$ instead of $f$? A function is not the same as one value of a function. $\endgroup$
    – YCor
    Commented Nov 19, 2023 at 13:53
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    $\begingroup$ I guess you mean, your question, not your inequality. I didn't vote closure, I imagine you can read who voted for closure and the question should be addressed to them, not to me (you didn't address my comment by the way). It is likely that these users believe that MathSE would be more appropriate. $\endgroup$
    – YCor
    Commented Nov 20, 2023 at 11:50
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    $\begingroup$ @pie anyway I didn't make any inquiry and am not the person to ask $\endgroup$
    – YCor
    Commented Nov 20, 2023 at 12:59
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    $\begingroup$ @Dragonboy I did not vote to close. However, MathOverflow is supposed to be for research-level math. I see two ways to interpret your question. (1) Provide any solution at all to the problem. (2) Provide a solution that a contestant could plausibly provide under contest conditions. It seems that (1) has already been answered, and (2) is not a research-type question because mathematical research is not conducted under contest conditions (and anyway, such a solution may not exist). $\endgroup$
    – Timothy Chow
    Commented Nov 20, 2023 at 17:52
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    $\begingroup$ @pie It's not the difficulty of the problem that is relevant, but whether the question being asked is a research question. To repeat myself, the OP appears to be asking for either (1) any solution at all or (2) a solution findable under contest conditions. (1) is already answered and the trouble with (2) is that mathematical research is not conducted under contest conditions. $\endgroup$
    – Timothy Chow
    Commented Nov 21, 2023 at 15:33

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