There are natural numbers: $a$, $b$, $c$.
$\begin{cases} ab+bc+ca+\frac32(a+b+c)=5015,\\ 2abc-a-b-c=6366 \end{cases} $
I need to find the minimum value of $a+b+c$. To my mind there's something connected with the derivative. I've already tried to find an equation for $a+b+c$, but I stuck right here. If we do some math in the second equation, we can get this one
$a+b+c=3183/abc$ Everything's great, but if we consider $a+b+c = l$ (for example), then we need to find a minimum of this function. But as far as I'm concerned this is a Hyperbolic function, which doesn't have extreme values. So there's my first confusion.
Moving on I decided to do something with a first equation. But, unfortunately, no matter how I rearrange my $a$, $b$ and $c$ I get some equations and can't get anything useful out of it.
And the last thing. If I get an equation for $a+b+c$ from the first equation then find a derivative and its zero value, I get $c = -b$.
Well, that perfectly concludes the problem. Could anyone help me get this one done?
Sorry for image confusion.