Austria 2014

The AM-GM inequality gives us: $$12x^2+yz=x^2+x^2+\cdots +x^2+yz\ge 13\sqrt[13]{x^{24}yz}$$ Applying this idea to the other two expressions then yields $$\begin{array}{rl}(12x^2+yz)\cdot (12y^2+xz)\cdot (12z^2+xy)& \ge 13^3\sqrt[13]{x^{24}yz\cdot y^{24}xz\cdot z^{24}xy}\\ & =13^3\sqrt[13]{x^{26}{y}^{26}{z}^{26}}\\ & =2197x^2{y}^2{z}^2>2014x^2{y}^2{z}^2(\text{since }{x}^2{y}^2{z}^2>0)\end{array}$$ The left-hand side is therefore always greater than the right-hand side. It therefore follows that no positive real numbers $x$, $y$, $z$ can exist that solve the equation. $\square$