[b]p1.[/b] Let $A = \{D,U,K,E\}$ and $B = \{M, A, T,H\}$. How many maps are there from $A$ to $B$?


[b]p2.[/b] The product of two positive integers $x$ and $y$ is equal to $3$ more than their sum. Find the sum of all possible $x$.


[b]p3.[/b] There is a bag with $1$ red ball and $1$ blue ball. Jung takes out a ball at random and replaces it with a red ball. Remy then draws a ball at random. Given that Remy drew a red ball, what is the probability that the ball Jung took was red?


[b]p4.[/b] Let $ABCDE$ be a regular pentagon and let $AD$ intersect $BE$ at $P$. Find $\angle APB$.


[b]p5.[/b] It is Justin and his $4\times 4\times 4$ cube again! Now he uses many colors to color all unit-cubes in a way such that two cubes on the same row or column must have different colors. What is the minimum number of colors that Justin needs in order to do so?


[b]p6.[/b] $f(x)$ is a polynomial of degree $3$ where $f(1) = f(2) = f(3) = 4$ and $f(-1) = 52$. Determine $f(0)$.


[b]p7.[/b] Mike and Cassie are partners for the Duke Problem Solving Team and they decide to meet between $1$ pm and $2$ pm. The one who arrives first will wait for the other for $10$ minutes, the lave. Assume they arrive at any time between $1$ pm and $2$ pm with uniform probability. Find the probability they meet.


[b]p8.[/b] The remainder of $2x^3 - 6x^2 + 3x + 5$ divided by $(x - 2)^2$ has the form $ax + b$. Find $ab$.


[b]p9.[/b] Find $m$ such that the decimal representation of m! ends with exactly $99$ zeros.


[b]p10.[/b] Let $1000 \le n = \overline{DUKE}  \le 9999$. be a positive integer whose digits $\overline{DUKE}$ satisfy the divisibility condition: $$1111 | \left( \overline{DUKE} + \overline{DU} \times \overline{KE} \right)$$ Determine the smallest possible value of $n$.


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. To determine the number of maps from set \( A \) to set \( B \), we need to consider the definition of a map (or function) in set theory. A map from \( A \) to \( B \) assigns each element in \( A \) to an element in \( B \).

2. Set \( A \) has 4 elements: \( \{D, U, K, E\} \).

3. Set \( B \) also has 4 elements: \( \{M, A, T, H\} \).

4. For each element in \( A \), there are 4 possible choices in \( B \). Therefore, the total number of maps from \( A \) to \( B \) is calculated by raising the number of choices to the power of the number of elements in \( A \):

\[
4^4 = 256
\]

The final answer is \(\boxed{256}\).