<s>[INST] Determine the irrational numbers $x$ such that both $x^2+2x$ and $x^3-6x$ are rational numbers I did not make any progress. The problem is from RMC 2008.
The only idea that I have is:
Try to find sets of irrational numbers such that every number in the set multiplied by another number in the set yields a rational number.
 [/INST] I add yet another solution hopefully a simple one.
Since $x^2+2x$ is rational so too is $x^2+2x+1=(x+1)^2$. 
Let $y=x+1$, so $y^2$ is rational,  then $x^3-6x=y(y^2-3)-3y^2+5$ is rational, which means that $y(y^2-3)$ is rational. 
Now since $y$ is irrational and $y^2-3$ rational we must have $y^2-3=0$, thus $y=\pm \sqrt{3}$ and $x=-1\pm\sqrt{3}$
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