Saudi Arabia Mathematical Competitions 2012

The inequality is equivalent to $$x^6-x^3+\frac{1}{4}+x^4-x^2+\frac{1}{4}+x^2-x+\frac{1}{4}>0,$$ which in turn is equivalent to $${\left(x^3-\frac{1}{2}\right)}^2+{\left(x^2-\frac{1}{2}\right)}^2+{\left(x-\frac{1}{2}\right)}^2>0.$$

Since $\frac{1}{\sqrt{2}}\ne \frac{1}{2}$, equality cannot occur.