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Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$

This question has been asked on MSE without receiving any answers. Just some comment by achille hui and myself.
I think this result is true because for $n=1$ we have $6(A^3+B^3+C^3)+1\ge 5(A^2+B^2+C^2)$
but unfortunately I didn't succeed to generalize for $n\ge2$.

N.B.: My apologize if this question is not suitable on math overflow.

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    $\begingroup$ Why do your expect this to be true? Have you compiled extensive evidence or is this posed somewhere as a problem? $\endgroup$ – Michael Renardy Sep 7 '14 at 16:03
  • $\begingroup$ @MichaelRenardy You can look at the link on MSE, it's a problem but achille hui was thinking that perhaps this problem is false, but as I said for $n=1$ is true because we can prove that $6(A^3+B^3+C^3)+I_n\ge 5(A^2+B^2+C^2)$ with proviso that $A+B+C=I_n$. $\endgroup$ – Krokop Sep 7 '14 at 16:09
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    $\begingroup$ @MichaelRenardy Sigh. That particular OP posts question after question from contests, mostly in China but not always. I continue to think that the best approach is to get the OP to get the answers from the contest organizers, which are typically published eventually. $\endgroup$ – Will Jagy Sep 7 '14 at 20:13
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    $\begingroup$ @ChristianRemling, sure. I find many contest problems fascinating. I do think people should be aware that this is a contest problem, someone knows the answer very well, and the person who originally asked on MSE does not care very much about it, there seems to be a bit of compulsion involved. Meanwhile, if I remain the only vote to close, there will be plenty of time for answers to appear here; also answers can be posted at MSE any time regardless. It is probably fair to say that I dislike the machine-gun rapid-fire postings more than I dislike contest problems. $\endgroup$ – Will Jagy Sep 7 '14 at 21:12
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    $\begingroup$ @WillJagy But what if that OP doesn't get access to the solutions? $\endgroup$ – crskhr May 30 '16 at 16:08

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