Problem source

For the real numbers $a_1$, $a_2$, $a_3$, $\ldots$,
\begin{questionparts}
\item prove that $a_1^2+a_2^2 \ge 2a_1a_2$,
\item prove that $a_1^2+a_2^2 +a_3^2  \ge a_2a_3 + a_3a_1 +a_1a_2$,
\item prove that $3(a_1^2+a_2^2 +a_3^2 +a_4^2)
 \ge 2(a_1a_2+a_1a_3 + a_1a_4 +a_2a_3 + a_2a_4 +a_3a_4)$,
\item state and prove a generalisation of (iii) to the case of $n$ real
numbers,
\item prove that 
$$
\left(\sum_{i=1}^n a_i \right)^2 \ge {2n\over n-1} \sum_{i,j} a_ia_j,
$$
where the latter sum is taken over all pairs $(i,j)$ with $1\le i<j\le
n$. 
\end{questionparts}