
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
<details><summary>D stands for Dedekind, Z stands for Zermelo</summary>they had two problem sets under those two names</details> 


<u>Set 1</u>**D1 / Z1.** What is  $1 + 2 \cdot 3$ ?**D2.** What is the average of the first  $9$  positive integers?**D3 / Z2.** A square of side length  $2$  is cut into  $4$  congruent squares. What is the perimeter of one of the  $4$  squares?**D4.** Find the ratio of a circle’s circumference squared to the area of the circle.**D5 / Z3.**  $6$  people split a bag of cookies such that they each get  $21$  cookies. Kyle comes and demands his share of cookies. If the  $7$  people then re-split the cookies equally, how many cookies does Kyle get?  


<u>Set 2</u>**D6.** How many prime numbers are perfect squares?**D7.** Josh has an unfair  $4$ -sided die numbered  $1$  through  $4$ . The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either  $1$  or  $3$ ?**D8.** If Alice consumes  $1000$  calories every day and burns  $500$  every night, how many days will it take for her to first reach a net gain of  $5000$  calories?**D9 / Z4.** Blobby flips  $4$  coins. What is the probability he sees at least one heads and one tails?**D10.** Lillian has  $n$  jars and  $48$  marbles. If George steals one jar from Lillian, she can fill each jar with  $8$  marbles. If George steals  $3$  jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill?


<u>Set 3</u>**D11 / Z6.** How many perfect squares less than  $100$  are odd?**D12.** Jash and Nash wash cars for cash. Jash gets  $\$ 6 $ for each car, while Nash gets $ \ $11$  per car. If Nash has earned  $\$ 1 $ more than Jash, what is the least amount of money that Nash could have earned?**D13 / Z5.** The product of $ 10 $ consecutive positive integers ends in $ 3 $ zeros. What is the minimum possible value of the smallest of the $ 10 $ integers?**D14 / Z7.** Guuce continually rolls a fair $ 6 $-sided dice until he rolls a $ 1 $ or a $ 6 $. He wins if he rolls a $ 6 $, and loses if he rolls a $ 1$. What is the probability that Guuce wins?**D15 / Z8.** The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square?


PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [here ](https://artofproblemsolving.com/community/c3h2916250p26045695)and Z.15-25  [here](https://artofproblemsolving.com/community/c3h2916258p26045774). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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