
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**p1.** Let  $A = \{D,U,K,E\}$  and  $B = \{M, A, T,H\}$ . How many maps are there from  $A$  to  $B$ ?**p2.** The product of two positive integers  $x$  and  $y$  is equal to  $3$  more than their sum. Find the sum of all possible  $x$ .**p3.** 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?**p4.** Let  $ABCDE$  be a regular pentagon and let  $AD$  intersect  $BE$  at  $P$ . Find  $\angle APB$ .**p5.** 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?**p6.**  $f(x)$  is a polynomial of degree  $3$  where  $f(1) = f(2) = f(3) = 4$  and  $f(-1) = 52$ . Determine  $f(0)$ .**p7.** 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.**p8.** The remainder of  $2x^3 - 6x^2 + 3x + 5$  divided by  $(x - 2)^2$  has the form  $ax + b$ . Find  $ab$ .**p9.** Find  $m$  such that the decimal representation of m! ends with exactly  $99$  zeros.**p10.** 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 [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Present the answer in LaTex format: \boxed{Your answer}
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