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1. Count positive factors of 1100 ●○○○○ 10 ptsHow many distinct positive factors does $1100$ have?
2. Odd reciprocals with terminating decimals ●●○○○ 20 ptsIf $\frac{a}{b}$ is the probability that the reciprocal of a randomly selected positive odd integer less than $2010$ gives a terminating decimal, with $a$ and $b$ relatively prime positive integers, find $a+b$.
3. 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)?
4. Smallest multiple beyond 2010 with terminating decimals ●●○○○ 20 ptsLet $S$ be the set of all integers $k$ such that $\frac{17k}{66}$ and $\frac{13k}{105}$ are both terminating decimals. What is the smallest integer in $S$ that is greater than 2010?
5. Divisors of exactly one number ●●○○○ 20 ptsHow many natural numbers divide exactly one of 1080 and 1800, but not both?
6. Smallest result from changing multiplications to divisions ●●●○○ 50 ptsIn the expression $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$, some of the "$\times$" signs are changed to "$\div$" so that the final result is still a natural number $N$. Find the smallest possible $N$.
7. Minimum distinct values in a ratio chain ●●●○○ 50 ptsLet $a_{1}, a_{2}, \ldots, a_{2006}$ be positive integers (not necessarily distinct) such that the 2005 ratios $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \ldots, \frac{a_{2005}}{a_{2006}}$ are all distinct. What is the least possible number of different values among $a_{1}, a_{2}, \ldots, a_{2006}$?
8. Least n with many unit-fraction solutions ●●●○○ 50 ptsLet $x$, $y$ and $n$ be positive integers, and count the solutions $(x, y)$ of $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ with $x \le y$. For $n=4$ there are exactly three such solutions. Find the least value of $n$ for which the number of solutions exceeds one thousand.
9. Initial terms of a recursive sequence ●●●○○ 50 ptsLet the sequence of positive integers $\{a_n\}$ satisfy $a_{n+3}=a_{n+2}\left(a_{n+1}+2a_n\right)$ for $n=1,2,\ldots$, and $a_6=2288$. Find $a_1$, $a_2$, $a_3$.
10. Natural numbers with terminating quotients ●●●○○ 50 ptsFind all natural numbers $n \geq 2$ such that for every positive integer $k < n$, the quotient $\frac{n}{k}$ has a finite decimal representation (or is a whole number). List the values in increasing order.

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