[b]p1.[/b] Find all $x$ such that $(\ln (x^4))^2 = (\ln (x))^6$.


[b]p2.[/b] 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?


[b]p3.[/b] 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 $n$.)


[b]p4.[/b] Let $F_n$ denote the $n^{th}$ Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. Find the last digit of $$\sum^{97!+4}_{i=0}F_i.$$


[b]p5.[/b] Find all prime numbers $p$ such that $2p + 1$ is a perfect cube.


[b]p6.[/b] 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?


[b]p7.[/b] $S$ 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?


[b]p8.[/b] 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.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png[/img]

[b]p9.[/b] Let $x > y > 0$ be real numbers. Find the minimum value of $\frac{x}{y} + \frac{4x}{x-y}$ .


[b]p10.[/b] 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.


[b]p11.[/b] Find the sum of the largest and smallest prime factors of $9^4 + 3^4 + 1$.


[b]p12.[/b] 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?


[b]p13.[/b] 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 [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
**p1.** Find all \( x \) such that \( (\ln (x^4))^2 = (\ln (x))^6 \).

1. Start by simplifying the left-hand side of the equation:
   \[
   (\ln (x^4))^2 = (\ln (x^4))^2 = (4 \ln (x))^2 = 16 (\ln (x))^2
   \]

2. Now, equate this to the right-hand side:
   \[
   16 (\ln (x))^2 = (\ln (x))^6
   \]

3. Let \( y = \ln (x) \). The equation becomes:
   \[
   16 y^2 = y^6
   \]

4. Rearrange the equation:
   \[
   y^6 - 16 y^2 = 0
   \]

5. Factor out \( y^2 \):
   \[
   y^2 (y^4 - 16) = 0
   \]

6. Solve for \( y \):
   \[
   y^2 = 0 \quad \text{or} \quad y^4 - 16 = 0
   \]

7. This gives:
   \[
   y = 0 \quad \text{or} \quad y^4 = 16
   \]

8. Solving \( y^4 = 16 \):
   \[
   y = \pm \sqrt[4]{16} = \pm 2
   \]

9. Therefore, \( y \) can be \( 0, 2, \) or \( -2 \).

10. Recall \( y = \ln (x) \):
    \[
    \ln (x) = 0 \quad \Rightarrow \quad x = e^0 = 1
    \]
    \[
    \ln (x) = 2 \quad \Rightarrow \quad x = e^2
    \]
    \[
    \ln (x) = -2 \quad \Rightarrow \quad x = e^{-2}
    \]

The final answer is \( \boxed{1, e^2, e^{-2}} \).