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1. 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 $\frac{1}{7}$?
2. Doubling map sequence term ●●○○○ 20 ptsA sequence $\{a_n\}$ satisfies $a_{n+1}=2a_n$ if $0 \leqslant a_n < \frac{1}{2}$ and $a_{n+1}=2a_n-1$ if $\frac{1}{2} \leqslant a_n < 1$, for every positive integer $n$, and $a_1=\frac{6}{7}$. Find $a_{24}$. Round to two decimal places.
3. Roots of a symmetric odd function ●●●○○ 50 ptsLet $y=f(x)$ be an odd function defined on $\mathbf{R}$ whose graph is symmetric about the line $x=1$. When $0<x\leq 1$, $f(x)=\log_3 x$. Find the sum of all real roots of the equation $f(x)=-\frac{1}{3}+f(0)$ in the interval $(0,10)$.
4. Sum of values of a shifted odd function ●●●○○ 50 ptsGiven that $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(1) = 1$. If the graph of $f(x)$ is translated one unit to the right, the resulting graph is that of an even function. Find the sum $f(1) + f(2) + f(3) + \ldots + f(2009)$.
5. Intersections of a log and a sine graph ●●●●○ 100 ptsHow many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?
6. Distinct values of a symmetric function ●●●●○ 100 ptsGiven a function $f$ for which $f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$ for all real $x$, what is the largest number of different values that can appear in the list $f(0), f(1), f(2), \ldots, f(999)$?