Question 3
Let $b_1,b_2,\ldots$ be a sequence of positive real numbers such that for each $n\ge1$,
$$b_{n+1}^2\ge\frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+\cdots+\frac{b_n^2}{n^3}.$$
Show that there is a positive integer $M$ such that
$$\sum_{n=1}^M\frac{b_{n+1}}{b_1+b_2+\cdots+b_n}>\frac{2013}{1013}.$$
$$b_{n+1}^2\ge\frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+\cdots+\frac{b_n^2}{n^3}.$$
Show that there is a positive integer $M$ such that
$$\sum_{n=1}^M\frac{b_{n+1}}{b_1+b_2+\cdots+b_n}>\frac{2013}{1013}.$$