
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.**Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs  $100$  pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs  $400$  pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds?**p2.** How many digits does the product  $2^{42}\cdot 5^{38}$  have?**p3.** Square  $ABCD$  has equilateral triangles drawn external to each side, as pictured. If each triangle is folded upwards to meet at a point  $E$ , then a square pyramid can be made. If the center of square  $ABCD$  is  $O$ , what is the measure of  $\angle OEA$ ?
![Image](https://cdn.artofproblemsolving.com/attachments/9/a/39c0096ace5b942a9d3be1eafe7aa7481fbb9f.png)**p4.** How many solutions  $(x, y)$  in the positive integers are there to  $3x + 7y = 1337$  ?**p5.** A trapezoid with height  $12$  has legs of length  $20$  and  $15$  and a larger base of length  $42$ . What are the possible lengths of the other base?**p6.** Let  $f(x) = 6x + 7$  and  $g(x) = 7x + 6$ . Find the value of a such that  $g^{-1}(f^{-1}(g(f(a)))) = 1$ .**p7.** Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between  $1:00$  and  $2:00$  this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between  $1:00$  and  $2:00$ . Silly Billy is impatient, though, and if he has to wait for Cindy, he will leave after  $15$  minutes. Cindy, on the other hand, will happily wait for Billy from whenever she arrives until  $2:00$ . What is the probability that Billy and Cindy will be able to dine together?**p8.** As pictured, lines are drawn from the vertices of a unit square to an opposite trisection point. If each triangle has legs with ratio  $3 : 1$ , what is the area of the shaded region?
![Image](https://cdn.artofproblemsolving.com/attachments/e/9/35a6340018edcddfcd7e085f8f6e56686a8e07.png)**p9.** For any positive integer  $n$ , let  $f_1(n)$  denote the sum of the squares of the digits of  $n$ . For  $k \ge 2$ , let  $f_k(n) = f_{k-1}(f_1(n))$ . Then,  $f_1(5) = 25$  and  $f_3(5) = f_2(25) = 85$ . Find  $f_{2012}(15)$ .**p10.** Given that  $2012022012$  has  $ 8$  distinct prime factors, find its largest prime factor.


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

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