[b]p1.[/b] Meghan Trainor is all about those base-systems. If she weighs $451$ pounds in some base, and $127$ pounds in a base that is twice as big, how much does she weigh in base $10$?


[b]p2.[/b] Taylor Swift made the song $15$ in $2008$, and the song $22$ in $2012$. If she had continued at the same rate, in what year would her song $1989$ have come out?


[b]p3.[/b] Sir Mix-A-Lot likes big butts and cannot lie. He is with his three friends, who all always lie. If each of the following statements was said by a different person, who is Sir Mix-A-Lot?
A: B and D both like big butts.
B: C and D either both like big butts, or neither of them do.
C: B does not like big butts.
D: A or C (or both) like big butts.


[b]p4.[/b] Mark Ronson is going to give some uptown funk-tions to you. Bruno Mars is at $1$ degree Fahrenheit right now (that is, time, in seconds, $t = 0$), and his heat h(t) is given by $h(t) = 2 \cdot h(t - 1) + t$. If he will explode when he is over $1,000,000$ degrees Fahrenheit, at what time t will Bruno Mars explode?


[b]p5.[/b] Jay-Z, who is a emcee, took $4$ AMC contests, for a total of $100$ problems. He got $99$ problems correct and $1$ problem wrong. His answer, $z$, to the problem he missed, was the square root of the correct $4$-digit answer, $j$. If $z$ is also $3$ times the sum of the digits of $j$, then what is the ordered tuple $(j, z)$?


PS. You had better use hide for answers.
### Problem 1:
1. Let the base in which Meghan weighs 451 pounds be \( b \). Then, in base \( b \), the number 451 represents:
   \[
   4b^2 + 5b + 1
   \]
2. In a base that is twice as big, \( 2b \), she weighs 127 pounds. In base \( 2b \), the number 127 represents:
   \[
   1(2b)^2 + 2(2b) + 7 = 4b^2 + 4b + 7
   \]
3. Since both expressions represent the same weight, we set them equal to each other:
   \[
   4b^2 + 5b + 1 = 4b^2 + 4b + 7
   \]
4. Simplify the equation:
   \[
   5b + 1 = 4b + 7 \implies b = 6
   \]
5. Substitute \( b = 6 \) back into the original base \( b \) expression to find the weight in base 10:
   \[
   4(6)^2 + 5(6) + 1 = 144 + 30 + 1 = 175
   \]

The final answer is \( \boxed{(2916, 54)} \)