$1)~$ Intelligent problem. Let $s:=4$ and $54=3(s^2+2t^2),t\geq0\implies t=1.$ It's well known that $x\in[s-2t,s-t],$ while $z\in[s+t,s+2t],$ from which the conclusion. The left equality for $(3,3,6)$ and the right equality for $(2,5,5).$ $2)~$ For example, $z^2-12z+45-xy=0.$ So $\frac\Delta4\geq0\implies xy\geq9.$ Equality for $(3,3,6).$ On the other hand, $yz\leq\frac{(y+z)^2}4=\frac{(12-x)^2}4\leq\frac{100}4=25.$ Equality for $(2,5,5).$

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