
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. Juan and Carlos draw the tadpoles in their design class (they are the ones with contour continuous). Which of the two has a greater area, if all the circles have the same radius and the equilateral triangle has a side equal to one diameter?
![Image](https://cdn.artofproblemsolving.com/attachments/7/5/b8cf02ea3222f95e4e96e095ceee39d80ade94.jpg)


p2. Find some positive  $N$  of  $10$  digits, all different from  $0$  and such that, by adding  $N$  with the product of its digits (which we will denote by  $P$ ), we obtain a number  $M$  that has a sum of digits equal to  $P$ .


p3. In each square of a  $ 3\times 3$  square one of the numbers  $1,2,3,...,9$   is written . It is known that the sum of the numbers of each is equal to  $15$ , and that the same happens with the numbers of each column or diagonal. $\bullet$  What number is in the center box? $\bullet$  Give examples of configurations that verify the properties.


p4. A mathematician tells another:
''I think you can guess how many grandchildren I have and what age are, if I tell you that the product of their ages is  $36$  and the sum of their ages coincides with the number of floors of the front building.'' 
His friend thinks for a while and replies:
''Your information is not enough, Caramba!.''
Then, the first adds smiling:
'' My older grandson is called Sandro .'' 
''Now it is another thing! '' 
says the second one, and gives the correct answer.
 What are the ages then?


p5. Calculate the area of the triangle  $ABC$ , knowing that  $BD = 4$ ,  $DE = 2$ ,  $EC = 6$ ,  $BF = FC = 3$ .
![Image](https://cdn.artofproblemsolving.com/attachments/6/0/11bbba5770b845380cc2fbe9f03847228752f9.jpg)


p6. Maca's family has  $4$  people: grandfather, mother, father, and herself. If we double the size of Maca's monthly scholarship, the family income will increase by  $5\%$ . If instead of the Maca scholarship, her mother's salary would be doubled, the family income will grow by  $15\%$ . The same procedure gives  $25\%$  in the case of Dad now.  In what percentage does the family income grow, if only the grandfather's salary is doubled?

PS. Juniors P6 was also [Seniors](https://artofproblemsolving.com/community/c4h2692089p23370183) P4.

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