[b]p6.[/b] Randy is playing a trivia game with $6$ questions. Each question has $3$ answer choices and if he answers all $6$ questions correctly, he wins $5000$ dollars. What is the expected amount of money Randy will win?


[b]p7.[/b] It has recently been proven that a sudoku puzzle requires at least $17$ numbers to be uniquely solvable. We section a $4\times 4$ grid of boxes into four $2\times 2$ squares. In each square we place the number $1,2,3$, or $4$. An arrangement is called [i]sudoku-like [/i] if there is exactly one of $1,2,3,4$ in each row, column, and $2\times 2$ box. How many sudoku-like arrangements are there? An example is given below.
[img]https://cdn.artofproblemsolving.com/attachments/6/a/f3c94ba7f064932c3789ac31880e2170b9a4e1.png[/img]


[b]p8.[/b] A [i]superperfect [/i] number is a number $n$ such that if $\sigma(n)$ denotes the sum of its factors, then $\sigma(\sigma(n)) = 2n$. For example, $\sigma(16) = 1+2+4+8+16 = 31$ and $\sigma(31) = 32$. $A$ and $B$ are distinct numbers such that they each have exactly $10$ factors and $|A - B| = 1$. Find the minimum possible value of $A +B$.


[b]p9.[/b] A trivia game awards up to $2018$ dollars split evenly among all of its winners such that each winner gets the maximum possible integer number of dollars. In a particular game if one more person had won each winner would have gotten two fewer dollars. How many possible number of winners are there for this game?


[b]p10.[/b] The number $0$ didn’t exist until $628$ AD, when it was introduced by the Indian mathematician Brahmagupta. The concept of $0$ did exist much before then.

Let $\oplus$ be a binary operator such that for any $3$ real numbers $a,b,c$, we have $$(a \oplus b)\oplus c = a \oplus (b \oplus c)$$ and $$a \oplus b \oplus c = 4abc -2ab -2bc -2ac + a +b +c. $$ Find all possible values of $20\oplus 18$.


PS. You should use hide for answers.
To solve this problem, we need to determine the number of possible winners \( n \) such that the difference in the amount of money each winner receives when there is one more winner is exactly 2 dollars.

1. Let \( n \) be the number of winners. Each winner receives \( \left\lfloor \frac{2018}{n} \right\rfloor \) dollars.
2. If there is one more winner, i.e., \( n+1 \) winners, each winner receives \( \left\lfloor \frac{2018}{n+1} \right\rfloor \) dollars.
3. According to the problem, the difference in the amount of money each winner receives when there is one more winner is 2 dollars:
   \[
   \left\lfloor \frac{2018}{n} \right\rfloor - \left\lfloor \frac{2018}{n+1} \right\rfloor = 2
   \]

4. We need to analyze the bounds of the floor functions:
   \[
   \frac{2018}{n} - 1 \leq \left\lfloor \frac{2018}{n} \right\rfloor \leq \frac{2018}{n}
   \]
   \[
   \frac{2018}{n+1} - 1 \leq \left\lfloor \frac{2018}{n+1} \right\rfloor \leq \frac{2018}{n+1}
   \]

5. Combining these inequalities, we get:
   \[
   1 \leq \frac{2018}{n} - \frac{2018}{n+1} < 4
   \]

6. Simplifying the difference:
   \[
   \frac{2018}{n} - \frac{2018}{n+1} = 2018 \left( \frac{1}{n} - \frac{1}{n+1} \right) = 2018 \left( \frac{(n+1) - n}{n(n+1)} \right) = \frac{2018}{n(n+1)}
   \]

7. Therefore, we have:
   \[
   1 \leq \frac{2018}{n(n+1)} < 4
   \]

8. Solving the inequality:
   \[
   1 \leq \frac{2018}{n(n+1)} \implies n(n+1) \leq 2018
   \]
   \[
   \frac{2018}{n(n+1)} < 4 \implies n(n+1) > \frac{2018}{4} = 504.5
   \]

9. Combining these results:
   \[
   504.5 < n(n+1) \leq 2018
   \]

10. We need to find integer values of \( n \) that satisfy this inequality. Testing values within the range:
    - For \( n = 22 \):
      \[
      22 \times 23 = 506 \quad (\text{valid})
      \]
    - For \( n = 44 \):
      \[
      44 \times 45 = 1980 \quad (\text{valid})
      \]

11. Checking all values between 22 and 44:
    - \( n = 23, 24, 25, \ldots, 43 \) all satisfy the inequality.

12. Therefore, the possible values of \( n \) are:
    \[
    \{22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44\}
    \]

13. Counting these values, we find there are 23 possible numbers of winners.

The final answer is \(\boxed{23}\)