Given real numbers $a,\,b,\,c$ such that
$a^2+ab+b^2=2$, $b^2+bc+c^2=3$, $c^2+ca+a^2=5$
Evaluate $ab+bc+ca$.
If nothing that we could do to algebraically manipulate this problem that comes to mind, as this is a hard problem, then what is left for us to be done, is, we can sum the three given equations, to get:
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