<s>[INST] Maximum value of $abc$ for $a, b, c > 0$ and $ab + bc + ca = 12$ $a,b,c$ are three positive real numbers such that $ab+bc+ca=12$.
Then find the maximum value of $abc$
 [/INST] I do not understand that. Same for any number can make this combination. 
equation:   $XY+XZ+YZ=N$  
Solutions in integers can be written by expanding the number of factorization: $N=ab$   
And vospolzovavschis solutions of Pell's equation: $p^2-(4k^2+1)s^2=1$  
$k$ - what some integer number given by us.  
Solutions can be written:  
$X=ap^2+2(ak+b+a)ps+(2(a-2b)k+2b+a)s^2$   
$Y=2(ak-b)ps+2(2ak^2+(a+2b)k+b)s^2$  
$Z=bp^2-2(2b+a)kps+(4bk^2-2ak-b)s^2$   
And more:  
$X=-2bp^2+2(k(4b+a)+b)ps-2((4b+2a)k^2+(2b-a)k)s^2$  
$Y=-(2b+a)p^2+2(k(4b+a)-b-a)ps-(8bk^2-(4b+2a)k+a)s^2$  
$Z=bp^2-2(2b+a)kps+(4bk^2-2ak-b)s^2$ 
Perhaps these formulas for someone too complicated. 
Then equation:  $XY+XZ+YZ=N$  
If we ask what ever number: $p$   
That the following sum can always be factored: $p^2+N=ks$ 
Solutions can be written.   
$X=p$   
$Y=s-p$ 
$Z=k-p$
8 - really is the maximum number of turns. But the smallest one can hardly get.
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