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1. Fifth term of Alicia's sequence ●○○○○ 10 ptsAlicia starts a sequence with $m=3$. Her algorithm is: 1. Write down the number $m$ as the first term. 2. If $m$ is even, set $n=\frac{1}{2}m$; if $m$ is odd, set $n=m+1$. 3. Write down the number $m+n+1$ as the next term. 4. Set $m$ equal to the value of the term just written in Step 3. 5. Repeat Steps 2, 3 and 4 until she has five terms, then stop. What is the fifth term of her sequence?
2. Iterated piecewise function value ●○○○○ 10 ptsFor integers $n$, let $f(n) = \begin{cases} n^2 + 1 & \text{if } n \text{ is odd}, \\ n^2 - 3n + 1 & \text{if } n \text{ is even}. \end{cases}$ Find $f(f(f(f(f(f(2))))))$.
3. Smallest non-negative signed sum ●●○○○ 20 ptsInsert $+$ or $-$ signs between the numbers $1, 2, 3, \ldots, 1989$, so that the sum is $\pm1\pm2\pm3\cdots\pm1989$ with the first number taken as $+$. What is the smallest non-negative value that the sum can take?
4. Locating a single bit error ●●○○○ 20 ptsA binary code $x_1x_2\ldots x_7$ satisfies the parity check equations $x_4\oplus x_5\oplus x_6\oplus x_7=0$, $x_2\oplus x_3\oplus x_6\oplus x_7=0$ and $x_1\oplus x_3\oplus x_5\oplus x_7=0$, where $\oplus$ is addition modulo 2. A code of this type was changed to $1101101$ after a bit error occurred at only one position $k$ during communication, so the bit at position $k$ flipped. Find $k$.
5. Power of a prime with prime shift ●●○○○ 20 ptsLet $p$ be a prime number such that $p + 3$ is also prime. Find $p^{11} - 52$.
6. Alternating-parity sum of reciprocal squares ●●●○○ 50 ptsCompute the value of $$\frac{1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots+\frac{1}{997^2}+\frac{1}{999^2}-\frac{1}{1002^2}-\frac{1}{1004^2}-\frac{1}{1006^2}-\cdots-\frac{1}{1998^2}-\frac{1}{2000^2}}{1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots+\frac{1}{999^2}+\frac{1}{1000^2}}$$ without using a calculator. Give your answer as a decimal.
7. Counting alternating-parity increasing sequences ●●●○○ 50 ptsA finite strictly increasing sequence of integers is called alternating if its first term is odd, its second term is even, its third term is odd, and so on, alternating in parity. The empty sequence also counts as alternating. Let $A(n)$ be the number of alternating sequences whose terms are all taken from $\{1,2,\ldots,n\}$, so $A(1)=2$ and $A(2)=3$. Find $A(20)$.
8. Possible second terms of a sequence ●●●●○ 100 ptsA sequence $a_1, a_2, \ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible?