Counterexample: $$$a=b=c=0.1$$$

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Not sure about the point of putting an MO-style inequality on codeforces, but just for the sake of completeness:

Note that by the rearrangement inequality, since $$$a^{3/2}, b^{3/2}, c^{3/2}$$$ and $$$\frac{1}{\sqrt{a}}, \frac{1}{\sqrt{b}}, \frac{1}{\sqrt{c}}$$$ are oppositely sorted, the LHS is at least $$$\frac{(a + b + c)^2}{3} \ge bc + ca + ab \ge 2(bc + ca + ab) - a^2 - b^2 - c^2$$$, where the last inequality also comes from the rearrangement inequality (or AM-GM) and we are done.

You could use Holder inequality to tackle the annoying sqrt part

then the LHS become ((a+b+c)^2)/3

Then you can use almost anything to prove.(AMGM, Muirhead, etc)

No need of this problem here

Nice inequality