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9 problems match “repeating decimals”. Clear

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1. Digit at a given position in 1/37 ●○○○○ 10 ptsFind the $291^{\text{st}}$ digit after the decimal point in the decimal expansion of $\dfrac{1}{37}$.
2. Ninth digit of 3/7 ●○○○○ 10 ptsFind the 9th digit to the right of the decimal point in the decimal representation of $\frac{3}{7}$.
3. Next term of a fraction sequence ●○○○○ 10 ptsFind the next term of the sequence $0, 0.5, 0.\overline{6}, 0.75, \dots$ and give it as a decimal. Round to three decimal places.
4. 1000th digit of 4/15 ●○○○○ 10 ptsWhat is the 1000th digit to the right of the decimal point when $\frac{4}{15}$ is expressed as a decimal?
5. 2007th digit of one seventh ●●○○○ 20 ptsWhat is the $2007^{\text{th}}$ digit to the right of the decimal point in the decimal expansion of $\frac{1}{7}$?
6. Repeating decimals of n/30 ●●○○○ 20 ptsFor how many integers $n$ from $1$ to $29$ is the fraction $\frac{n}{30}$ a repeating decimal (that is, its decimal expansion does not terminate)?
7. Repeating block length for 1/17 ●●○○○ 20 ptsWhat is the number of digits in the repeating block of the decimal expansion of $\frac{1}{17}$?
8. Digit sum of a sum of fractions ●●○○○ 20 ptsWhat is the value of $a+b+c+d+e+f$, where the decimal representation of $\frac{4}{37}+\frac{3}{11}+\frac{23}{9}$ is $2.abcdef\ldots$?
9. 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 $X$.