Putnam 1969

Let $D_n$ be the determinant of order $n$ of which the element in the $i$th row and the $j$th column is the absolute value of the difference of $i$ and $j$. Find the value of $D_n$.

The terms of a sequence $T_n$ satisfy $$T_n T_{n+1} = n (n = 1, 2, 3, \dots) \text{ and } \lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1.$$ Find the value of $\pi T_1^2$.

Let $a_1 < a_2 < a_3 < \cdots$ be an increasing sequence of positive integers. Let the series $$\sum_{n=1}^{\infty} \frac{1}{a_n}$$ be convergent. For any number $x$, let $k(x)$ be the number of the $a_n$'s which do not exceed $x$. Find the value of \lim_{x \to \infty} \frac{k(x)}{x}.