
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. In the month of January of a certain year, there were exactly four Mondays and four Fridays. What day of the week was February  $ 1$  ?


p2. Determine all positive integers  $n$  that have the following property: "Among the positive divisors of   $n$  different than  $ 1$  and  $n$ , the largest is  $35$  times the smallest."


p3. Vecindad Island has  $2011$  inhabitants, divided into three types: the innocent, the wicked and the capricious, the innocent always tell the truth, the wicked always lie, and the capricious alternately tell lies one day and truths the next. A reporter visits the island for two days. 
On the first day, the reporter interviews all the inhabitants: $\bullet$  The first one says: "There is exactly one villain on the island." $\bullet$  The second says: "There are exactly two villains on the island." $\bullet$  And so on until he reaches inhabitant number  $2011$ , who says: "There are exactly  $2011$  villains on the island."
On the second day, the reporter interviews everyone again in the same order: $\bullet$  The first says: "There is exactly one innocent on the island." $\bullet$  The second says: "There are exactly two innocents on the island." $\bullet$  And so on until he reaches inhabitant number  $2011$ , who says: "There are exactly  $2011$  innocents on the island."
How many capricious are there on the island?


p4. Let  $ABCDE$  be a regular pentagon such that the star  $ACEBD$  has area  $ 1$ . Let  $P$  be the intersection point of  $AC$  and  $BE$ , let  $Q$  be the intersection point of  $BD$  and  $CE$ . Determine the area of quadrilateral APQD.
![Image](https://cdn.artofproblemsolving.com/attachments/0/3/dcb0609bc85699b600b61ed97f6345a6b2b832.png)


p5. Determine all positive integers  $a, b, c, d$  with  $a <b <c <d$ , such that  $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d}$  an integer.

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