<s>[INST] Finding the number of real roots of an unusual(!) equation How many real roots does the below equation have?
\begin{equation*}
\frac{x^{2000}}{2001}+2\sqrt{3}x^2-2\sqrt{5}x+\sqrt{3}=0
\end{equation*}
A) 0  B) 11 C) 12 D) 1 E) None of these
I could not come up with anything.
(Turkish Math Olympiads 2001)
 [/INST] We have that $x^{2000} \geq 0$, because squares are nonnegative.
Further, we have $(x-\frac{1}{2}\sqrt{\frac{5}{3}})^2 \geq 0$. This gives $x^2-\sqrt{\frac{5}{3}}x+\frac{5}{12} \geq 0$
Therefore $2 \sqrt{3}x^2-2\sqrt{5}x+\frac{10}{12}\sqrt{3} \geq 0$
Therefore $2 \sqrt{3}x^2-2\sqrt{5}x+\sqrt{3} > 0$
Therefore $\frac{x^{2000}}{2001} + 2 \sqrt{3}x^2-2\sqrt{5}x+\sqrt{3} > 0$
Therefore there are no real roots. 
</s>