
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>B stands for Bernoulli, G stands for Germain</summary>they had two problem sets under those two names</details> 

<u>Set 1</u>**B1 / G1** Find  $20^3 + 2^2 + 3^1$ .**B2** A piece of string of length  $10$  is cut  $4$  times into strings of equal length. What is the length of each small piece of string?**B3 / G2** What is the smallest perfect square that is also a perfect cube?**B4** What is the probability a  $5$ -sided die with sides labeled from  $1$  through  $5$  rolls an odd number?**B5 / G3** Hanfei spent  $14$  dollars on chicken nuggets at McDonalds.  $4$  nuggets cost  $3$  dollars,  $6$  nuggets cost  $4$  dollars, and  $12$  nuggets cost  $9$  dollars. How many chicken nuggets did Hanfei buy?


<u>Set 2</u>**B6** What is the probability a randomly chosen positive integer less than or equal to  $15$  is prime?**B7** Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled  $1$  through  $6$ . What is the probability that the sum is greater than  $5$ ?**B8 / G4** What is the radius of a circle with area  $4$ ?**B9** What is the maximum number of equilateral triangles on a piece of paper that can share the same corner?**B10 / G5** Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in  $20$  minutes. Bab can make a pizza in  $30$  minutes. If Bob and Bab want to make  $50$  pizzas in total, how many hours would that take them?


<u>Set 3</u>**B11 / G6** Find the area of an equilateral rectangle with perimeter  $20$ .**B12 / G7** What is the minimum possible number of divisors that the sum of two prime numbers greater than  $2$  can have?**B13 / G8** Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard  rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins?**B14 / G9** Aven has  $4$  distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table?**B15 / G10** Find the largest  $7$ -digit palindrome that is divisible by  $11$ .


PS. You should use hide for answers. Rest problems have been posted [here](https://artofproblemsolving.com/community/c3h3132170p28376644). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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