
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.** I open my  $2010$ -page dictionary, whose pages are numbered  $ 1$  to  $2010$  starting on page  $ 1$  on the right side of the spine when opened, and ending with page  $2010$  on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of  $6$ ?**p2.** Let  $A$  be the number of positive integer factors of  $128$ .
Let  $B$  be the sum of the distinct prime factors of  $135$ .
Let  $C$  be the units’ digit of  $381$ .
Let  $D$  be the number of zeroes at the end of  $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$ .
Let  $E$  be the largest prime factor of  $999$ .
Compute  $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$ .**p3.** The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of  $17$  and  $7$ .**p4.** A regular hexagon  $ABCDEF$  has area  $1$ . The sides $ AB$ ,  $CD$ , and  $EF$  are extended to form a larger polygon with  $ABCDEF$  in the interior. Find the area of this larger polygon.**p5.** For real numbers  $x$ , let  $\lfloor x \rfloor$  denote the greatest integer less than or equal to  $x$ . For example,  $\lfloor 3\rfloor  = 3$  and  $\lfloor 5.2 \rfloor = 5$ . Evaluate  $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$ .**p6.** The mean of five positive integers is  $7$ , the median is  $8$ , and the unique mode is  $9$ . How many possible sets of integers could this describe?**p7.** How many three digit numbers x are there such that  $x + 1$  is divisible by  $11$ ?**p8.** Rectangle  $ABCD$  is such that  $AD = 10$  and  $AB > 10$ . Semicircles are drawn with diameters  $AD$  and  $BC$  such that the semicircles lie completely inside rectangle  $ABCD$ . If the area of the region inside  $ABCD$  but outside both semicircles is  $100$ , determine the shortest possible distance between a point  $X$  on semicircle  $AD$  and  $Y$  on semicircle  $BC$ .**p9.**  $ 8$  distinct points are in the plane such that five of them lie on a line  $\ell$ , and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on  $\ell$ . How many triangles can be formed using some three of the  $ 8$  points?**p10.** Carl has  $10$  Art of Problem Solving books, all exactly the same size, but only  $9$  spaces in his bookshelf. At the beginning, there are  $9$  books in his bookshelf, ordered in the following way. $A - B - C - D - E - F - G - H - I$ He is holding the tenth book,  $J$ , in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out  $A$ , put  $J$  in its place, then take out  $D$ , put  $A$  in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is  $G$ . At the end, he is holding G in his hand, and his bookshelf looks like this. $C - I - H - J - F - B - E - D - A$ Give the order (start to finish) in which Carl took out the books, expressed as a  $9$ -letter string (word).


PS. You had better use hide for answers.

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