Category Archives: Algebra

On the $X$-$Y$ plane, two circles centered at $(0,0)$ with radius $1$ and $2$ respectively. Let point $A=(-1,0)$, $B=(1,0)$, and $C$ is a point on the bigger circle. Find the locus of the orthocenter $P$ of $\triangle{ABC}$. Click here for … Continue reading →

If $x=\sqrt[3]{9}+\sqrt[3]{3}+1$, find the value of $(\dfrac{2}{x}+1\big{)}^3$. Click here for the solution. Solution: Let $y=\sqrt[3]{3}$, we have $$x=y^2+y+1=\dfrac{y^3-1}{y-1}=\dfrac{(\sqrt[3]{3})^3-1}{\sqrt[3]{3}-1}=\dfrac{2}{\sqrt[3]{3}-1}$$ Therefore $$(\dfrac{2}{x}+1)^3=(2\cdot \dfrac{\sqrt[3]{3}-1}{2}+1)^3=(\sqrt[3]{3})^3=\boxed{3}$$