Hong Kong Preliminary Selection Contest

Let $x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$. Using the formula $(a+b{)}^3=a^3+b^3+3ab(a+b)$, we have $$\begin{gathered} x^3=(\sqrt{5}+2)+(\sqrt{5}-2)+3\sqrt[3]{(\sqrt{5}+2)(\sqrt{5}-2)}x \\ =2\sqrt{5}+3x. \end{gathered}$$ Rewrite this as $(x-\sqrt{5})(x^2+\sqrt{5}x+2)=0$. Since the quadratic equation $x^2+\sqrt{5}x+2=0$ has no real roots, we must have $x=\sqrt{5}$, and hence $$\left[{\left(\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}\right)}^{2014}\right]=[{\sqrt{5}}^{2014}]=5^{1007}.$$ Since the last three digits of powers of 5 have the pattern 005, 025, 125, 625, 125, 625, ..., it follows that the answer is 125.