
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.** Suppose that  $a \oplus b = ab - a - b$ . Find the value of   $$ ((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5. $$  **p2.** Neethin scores a  $59$  on his number theory test. He proceeds to score a  $17$ ,  $23$ , and  $34$  on the next three tests. What score must he achieve on his next test to earn an overall average of  $60$  across all five tests?**p3.** Consider a triangle with side lengths  $28$  and  $39$ . Find the number of possible integer lengths of the third side.**p4.** Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of?**p5.** Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after  $6$  minutes?**p6.** Charlotte writes all the positive integers  $n$  that leave a remainder of  $2$  when  $2018$  is divided by  $n$ . What is the sum of the numbers that she writes?**p7.** Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach  $(5, 5)$ ?
![Image](https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png)**p8.** Let  $a, b, c$  be positive numbers where  $a^2 + b^2 + c^2 = 63$  and  $2a + 3b + 6c = 21\sqrt7$ . Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $ .**p9.** What is the sum of the distinct prime factors of  $12^5 + 12^4 + 1$ ?**p10.** Allen starts writing all permutations of the numbers  $1$ ,  $2$ ,  $3$ ,  $4$ ,  $5$ ,  $6$   $7$ ,  $8$ ,  $9$ ,  $10$  on a blackboard. At one point he writes the permutation  $9$ ,  $4$ ,  $3$ ,  $1$ ,  $2$ ,  $5$ ,  $6$ ,  $7$ ,  $8$ ,  $10$ . David points at the permutation and observes that for any two consecutive integers  $i$  and  $i+1$ , all integers that appear in between these two integers in the permutation are all less than  $i$ . For example,  $4$  and  $5$  have only the numbers  $3$ ,  $1$ ,  $2$  in between them. How many of the  $10!$  permutations on the board satisfy this property that David observes?**p11.** (Estimation) How many positive integers less than  $2018$  can be expressed as the sum of  $3$  square numbers?


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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