<reponame>ineq-tech/inequality
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\begin{document}
	
	\section{SgMO}\label{sec:sgmo}
	
		\begin{problem}[Singapore Team Selection Test $2009$, problem $2$]
			If $a,b,c$ are three positive real numbers such that $ab+bc+ca=1$, prove that
				\begin{align*}
					\sqrt[3]{\dfrac{1}{a}+6b}+\sqrt[3]{\dfrac{1}{b}+6c}+\sqrt[3]{\dfrac{1}{c}+6a}
						& \leq\dfrac{1}{abc}
				\end{align*}
		\end{problem}
		
		\begin{problem}[Singapore Team Selection Test $2008$, problem $2$]
			Let $x_{1},\ldots,x_{n}$ be positive real numbers such that $x_{1}\cdots x_{n}=1$. Prove that
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{1}{n-1+x_{i}}
						& \leq1
				\end{align*}
		\end{problem}
		
		\begin{problem}[Singapore Team Selection Test $2007$, problem $2$]
			Prove the inequality
				\begin{align*}
					\sum_{i<j}\dfrac{a_{i}a_{j}}{a_{i}+a_{j}}
						& \leq \dfrac{n}{2(a_{1}+\ldots+a_{n})}\sum_{i<j}a_{i}a_{j}
				\end{align*}
			for all positive real numbers $a_{1},\ldots,a_{n}$.
		\end{problem}
		
		\begin{problem}[Singapore $2006$, problem $2$]
			Let $n>1$ be an integer and $x_{1},\ldots,x_{n}$ be real numbers such that
				\begin{align*}
					|x_{1}|+\ldots+|x_{n}|
						& = 1\quad\mbox{and}\\
					x_{1}+\ldots+x_{n}
						& = 0
				\end{align*}
			Prove that
				\begin{align*}
					\left|\dfrac{x_{1}}{1}+\ldots+\dfrac{x_{n}}{n}\right|
						& \leq \dfrac{1}{2}\left(1-\dfrac{1}{n}\right)
				\end{align*}
		\end{problem}
		
		\begin{problem}[Singapore $2004$, problem $2$]
			Let $a,b,c$ be real numbers such that $ab+bc+ca=1$ and $0<a,b,c<1$. Prove that
				\begin{align*}
					\dfrac{a}{1-a^{2}}+\dfrac{b}{1-b^{2}}+\dfrac{c}{1-c^{2}}
						& \geq\dfrac{3\sqrt{3}}{2}
				\end{align*}
		\end{problem}
\end{document}