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1. 101st digit of a period-200 reciprocal ●●●○○ 50 ptsLet $p$ be a prime number such that the period of the decimal expansion of $\frac{1}{p}$ is 200. That is, $\frac{1}{p}=0.XXXX\ldots$ for some block $X$ of 200 digits, but $\frac{1}{p}\neq 0.YYYY\ldots$ for every block $Y$ with fewer than 200 digits. Find the 101st digit, counting from the left, of $X$.
2. Longest reciprocal cycle below 100 ●●●○○ 50 ptsFind the positive integer $d < 100$ for which the decimal expansion of $\frac{1}{d}$ has the longest recurring cycle.
3. Longest reciprocal cycle below 2000 ●●●●○ 100 ptsFind the value of $d < 2000$ for which $\frac{1}{d}$ has the longest recurring cycle in its decimal fraction part.
4. Longest reciprocal cycle below 2500 ●●●●○ 100 ptsFind the value of $d < 2500$ for which $\frac{1}{d}$ has the longest recurring cycle in its decimal fraction part.