
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  $a$  be an integer. How many fractions  $\frac{a}{100}$  are greater than  $\frac17$  and less than  $\frac13$  ?.**p2.** Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were  $5$  different kinds of fish,  $3$  different rice colors, and  $11$  different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is  $6/11$ , then how many non-spicy sauces were there?**p3.** A palindrome is any number that reads the same forward and backward (for example,  $99$  and  $50505$  are palindromes but  $2020$  is not). Find the sum of all three-digit palindromes whose tens digit is  $5$ .**p4.** Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has  $64$  multiple choice questions with  $4$  choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a  $100\%$  can be expressed as  $2^{2^k}$ . Find  $k$ .**p5.** Consider a three-way Venn Diagram composed of three circles of radius  $1$ . The area of the entire Venn Diagram is of the form  $\frac{a}{b}\pi +\sqrt{c}$  for positive integers  $a$ ,  $b$ ,  $c$  where  $a$ ,  $b$  are relatively prime. Find  $a+b+c$ . (Each of the circles passes through the center of the other two circles)**p6.** The sum of two four-digit numbers is  $11044$ . None of the digits are repeated and none of the digits are  $0$ s. Eight of the digits from  $1-9$  are represented in these two numbers. Which one is not?**p7.** Al wants to buy cookies. He can buy cookies in packs of  $13$ ,  $15$ , or  $17$ . What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size?**p8.** Let  $\vartriangle ABC$  be a right triangle with base  $AB = 2$  and hypotenuse  $AC = 4$  and let  $AD$  be a median of  $\vartriangle ABC$ . Now, let  $BE$  be an altitude in  $\vartriangle ABD$  and let  $DF$  be an altitude in  $\vartriangle ADC$ . The quantity  $(BE)^2 - (DF)^2$  can be expressed as a common fraction  $\frac{a}{b}$  in lowest terms. Find  $a + b$ .**p9.** Let  $P(x)$  be a monic cubic polynomial with roots  $r$ ,  $s$ ,  $t$ , where  $t$  is real. Suppose that  $r + s + 2t = 8$ ,  $2rs + rt + st = 12$  and  $rst = 9$ . Find  $|P(2)|$ .**p10.** Let S be the set  $\{1, 2,..., 21\}$ . How many  $11$ -element subsets  $T$  of  $S$  are there such that there does not exist two distinct elements of  $T$  such that one divides the other?


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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