, 2014

The AM-GM inequality gives us:
$$12x^2 + yz = x^2 + x^2 + \dots + x^2 + yz \ge 13 \sqrt[13]{x^{24}yz}$$
Applying this idea to the other two expressions then yields
$$\begin{aligned} (12x^2 + yz) \cdot (12y^2 + xz) \cdot (12z^2 + xy) &\ge 13^3 \sqrt[13]{x^{24}yz \cdot y^{24}xz \cdot z^{24}xy} \\ &= 13^3 \sqrt[13]{x^{26}y^{26}z^{26}} \\ &= 2197x^2y^2z^2 > 2014x^2y^2z^2 \quad (\text{since } x^2y^2z^2 > 0) \end{aligned}$$
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. $\square$