[u]Round 1[/u]


[b]p1.[/b] Elaine creates a sequence of positive integers $\{s_n\}$. She starts with $s_1 = 2018$. For $n \ge 2$, she sets $s_n =\frac12 s_{n-1}$ if $s_{n-1}$ is even and $s_n = s_{n-1} + 1$ if $s_{n-1}$ is odd. Find the smallest positive integer $n$ such that $s_n = 1$, or submit “$0$” as your answer if no such $n$ exists.


[b]p2.[/b] Alice rolls a fair six-sided die with the numbers $1$ through $6$, and Bob rolls a fair eight-sided die with the numbers $1$ through $8$. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins?


[b]p3.[/b] Four circles each of radius $\frac14$ are centered at the points $\left( \pm \frac14, \pm \frac14 \right)$, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle?


[u]Round 2 [/u]


[b]p4.[/b] If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing?


[b]p5.[/b] Let $a_1 = 2018$, and for $n \ge 2$ define $a_n = 2018^{a_{n-1}}$ . What is the ones digit of $a_{2018}$?


[b]p6.[/b] We can write $(x + 35)^n =\sum_{i=0}^n c_ix^i$ for some positive integer $n$ and real numbers $c_i$. If $c_0 = c_2$, what is $n$?


[u]Round 3[/u]


[b]p7.[/b] How many positive integers are factors of $12!$ but not of $(7!)^2$?


[b]p8.[/b] How many ordered pairs $(f(x), g(x))$ of polynomials of degree at least $1$ with integer coefficients satisfy $f(x)g(x) = 50x^6 - 3200$?


[b]p9.[/b] On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between $1$ and $10$ (inclusive). What is the probability that the average of their three scores is an integer?


[u]Round 4[/u]


[b]p10.[/b] Find the largest positive integer N such that $$(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)$$ is divisible by $N$ for all choices of positive integers $a > b > c > d > e$.


[b]p11.[/b] Let $ABCDE$ be a square pyramid with $ABCD$ a square and E the apex of the pyramid. Each side length of $ABCDE$ is $6$. Let $ABCDD'C'B'A'$ be a cube, where $AA'$, $BB'$, $CC'$, $DD'$ are edges of the cube. Andy the ant is on the surface of $EABCDD'C'B'A'$ at the center of triangle $ABE$ (call this point $G$) and wants to crawl on the surface of the cube to $D'$. What is the length the shortest path from $G$ to $D'$? Write your answer in the form $\sqrt{a + b\sqrt3}$, where $a$ and $b$ are positive integers.


[b]p12.[/b] A six-digit palindrome is a positive integer between $100, 000$ and $999, 999$ (inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there?


PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. **Case 1: Alice rolls a $1$.**
   - No matter what Bob rolls, Alice will win. Hence, there are $8$ possible ways for Alice to win.

2. **Case 2: Alice rolls a $2$.**
   - If Bob rolls a $2$, $4$, $6$, or $8$, then Alice will win. Hence, there are $4$ possible ways for Alice to win.

3. **Case 3: Alice rolls a $3$.**
   - If Bob rolls a $3$ or $6$, then Alice will win. Hence, there are $2$ possible ways for Alice to win.

4. **Case 4: Alice rolls a $4$.**
   - If Bob rolls a $4$ or $8$, then Alice will win. Hence, there are $2$ possible ways for Alice to win.

5. **Case 5: Alice rolls a $5$.**
   - Alice will win if Bob rolls a $5$. Hence, there is $1$ possible way for Alice to win.

6. **Case 6: Alice rolls a $6$.**
   - Alice will win if Bob rolls a $6$. Hence, there is $1$ possible way for Alice to win.

Summarizing the cases, we have:
- $8$ ways for Alice to win when she rolls a $1$.
- $4$ ways for Alice to win when she rolls a $2$.
- $2$ ways for Alice to win when she rolls a $3$.
- $2$ ways for Alice to win when she rolls a $4$.
- $1$ way for Alice to win when she rolls a $5$.
- $1$ way for Alice to win when she rolls a $6$.

Adding these up, the total number of ways Alice can win is:
\[ 8 + 4 + 2 + 2 + 1 + 1 = 18 \]

The total number of possible outcomes when both Alice and Bob roll their dice is:
\[ 6 \times 8 = 48 \]

Therefore, the probability that Alice wins is:
\[ \frac{18}{48} = \frac{3}{8} \]

The final answer is \(\boxed{\frac{3}{8}}\)