
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.** On SeaBay, green herring costs  $\$ 2.50 $ per pound, blue herring costs $ \ $4.00$  per pound, and red herring costs  $\$ 5,85 $ per pound. What must Farmer James pay for $ 12 $ pounds of green herring and $ 7 $ pounds of blue herring, in dollars?**p2.** A triangle has side lengths $ 3 $, $ 4 $, and $ 6 $. A second triangle, similar to the first one, has one side of length $ 12 $. Find the sum of all possible lengths of the second triangle's longest side.**p3.** Hen Hao runs two laps around a track. Her overall average speed for the two laps was $ 20\% $ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap?**p4.** Square $ ABCD $ has side length $ 2 $. Circle $ \omega $ is centered at $ A $ with radius $ 2 $, and intersects line $ AD $ at distinct points $ D $ and $ E $. Let $ X $ be the intersection of segments $ EC $ and $ AB $, and let $ Y $ be the intersection of the minor arc $ DB $ with segment $ EC $. Compute the length of $ XY $ .**p5.** Hen Hao rolls $ 4 $ tetrahedral dice with faces labeled $ 1 $, $ 2 $, $ 3 $, and $ 4 $, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square.**p6.** Let $ N \ge 11 $ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $ 1 $ to $ N $, inclusive. Later, during the live drawing, another group of six numbers from $ 1 $ to $ N $ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $ N $?**p7.** An egg plant is a hollow cylinder of negligible thickness with radius $ 2 $ and height $ h $. Inside the egg plant, there is enough space for four solid spherical eggs of radius $ 1 $. What is the minimum possible value for $ h $?**p8.** Let $ a_1, a_2, a_3, ... $ be a geometric sequence of positive reals such that $ a_1 < 1 $ and $ (a_{20})^{20} = (a_{18})^{18} $. What is the smallest positive integer n such that the product $ a_1a_2a_3...a_n $ is greater than $ 1 $?**p9.** In parallelogram $ ABCD $, the angle bisector of $ \angle DAB $ meets segment $ BC $ at $ E $, and $ AE $ and $ BD $ intersect at $ P $. Given that $ AB = 9 $, $ AE = 16 $, and $ EP = EC $, find $ BC $.**p10.** Farmer James places the numbers $ 1, 2,..., 9 $ in a $ 3\times 3 $ grid such that each number appears exactly once in the grid. Let $ x_i $ be the product of the numbers in row $ i $, and $ y_i $ be the product of the numbers in column $ i $. Given that the unordered sets $ \{x_1, x_2, x_3\} $ and $ \{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made?


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

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