IMC 1996 Problem 4

Let $a_{1}=1$ , $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$ . Show that i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$ ; ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$