
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.** Find all  $x$  such that  $(\ln (x^4))^2 = (\ln (x))^6$ .**p2.** On a piece of paper, Alan has written a number  $N$  between  $0$  and  $2010$ , inclusive. Yiwen attempts to guess it in the following manner: she can send Alan a positive number  $M$ , which Alan will attempt to subtract from his own number, which we will call  $N$ . If  $M$  is less than or equal  $N$ , then he will erase  $N$  and replace it with  $N -M$ . Otherwise, Alan will tell Yiwen that  $M > N$ . What is the minimum number of attempts that Yiwen must make in order to determine uniquely what number Alan started with?**p3.** How many positive integers between  $1$  and  $50$  have at least  $4$  distinct positive integer divisors? (Remember that both  $1$  and  $n$  are divisors of  $n$ .)**p4.** Let  $F_n$  denote the  $n^{th}$  Fibonacci number, with  $F_0 = 0$  and  $F_1 = 1$ . Find the last digit of   $$ \sum^{97!+4}_{i=0}F_i. $$  **p5.** Find all prime numbers  $p$  such that  $2p + 1$  is a perfect cube.**p6.** What is the maximum number of knights that can be placed on a  $9\times 9$  chessboard such that no two knights attack each other?**p7.**  $S$  is a set of  $9$  consecutive positive integers such that the sum of the squares of the  $5$  smallest integers in the set is the sum of the squares of the remaining  $4$ . What is the sum of all  $9$  integers?**p8.** In the following infinite array, each row is an arithmetic sequence, and each column is a geometric sequence. Find the sum of the infinite sequence of entries along the main diagonal.
![Image](https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png)**p9.** Let  $x > y > 0$  be real numbers. Find the minimum value of  $\frac{x}{y} + \frac{4x}{x-y}$  .**p10.** A regular pentagon  $P = A_1A_2A_3A_4A_5$  and a square  $S = B_1B_2B_3B_4$  are both inscribed in the unit circle. For a given pentagon  $P$  and square  $S$ , let  $f(P, S)$  be the minimum length of the minor arcs  $A_iB_j$  , for  $1 \le i \le 5$  and  $1 \le j \le4$ . Find the maximum of  $f(P, S)$  over all pairs of shapes.**p11.** Find the sum of the largest and smallest prime factors of  $9^4 + 3^4 + 1$ .**p12.** A transmitter is sending a message consisting of  $4$  binary digits (either ones or zeros) to a receiver. Unfortunately, the transmitter makes errors: for each digit in the message, the probability that the transmitter sends the correct digit to the receiver is only  $80\%$ . (Errors are independent across all digits.) To avoid errors, the receiver only accepts a message if the sum of the first three digits equals the last digit modulo  $2$ . If the receiver accepts a message, what is the probability that the message was correct?**p13.** Find the integer  $N$  such that   $$ \prod^{8}_{i=0}\sec \left( \frac{\pi}{9}2^i \right)= N. $$  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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