
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. One day, a researcher placed two groups of species that were different, namely amoeba and bacteria in the same medium, each in a certain amount (in unit cells). The researcher observed that on the next day, which is the second day, it turns out that every cell species divide into two cells. On the same day every cell amoeba prey on exactly one bacterial cell. The next observation carried out every day shows the same pattern, that is, each cell species divides into two cells and then each cell amoeba prey on exactly one bacterial cell. Observation on day  $100$  shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day.


p2. It is known that  $n$  is a positive integer. Let  $f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$ .
Find  $f(13) + f(14) + f(15) + ...+ f(112).$ p3. Budi arranges fourteen balls, each with a radius of  $10$  cm. The first nine balls are placed on the table so that
form a square and touch each other. The next four balls placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four balls. If Bambang has fifty five balls each also has a radius of  $10$  cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang.


p4. Given a triangle  $ABC$  whose sides are  $5$  cm,  $ 8$  cm, and  $\sqrt{41}$  cm. Find the maximum possible area of ​​the rectangle can be made in the triangle  $ABC$ .


p5. There are  $12$  people waiting in line to buy tickets to a show with the price of one ticket is   $5,000.00$  Rp.. Known  $5$  of them they only have   $10,000$  Rp. in banknotes and the rest is only has a banknote of   $5,000.00$  Rp. If the ticket seller initially only has   $5,000.00$  Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?

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