[u]Set 1[/u]


[b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?


[b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?


[b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules?


PS. You should use hide for answers.
### Problem 1
1. Let \( d \) be the distance between Anastasia and Bananastasia.
2. Let \( t_A \) be the time it takes for Anastasia's yell to reach you, and \( t_B \) be the time it takes for Bananastasia's yell to reach you.
3. Given that you hear Anastasia’s yell 5 seconds before Bananastasia’s yell, we have:
   \[
   t_B = t_A + 5
   \]
4. Since the speed of sound is \( 343 \) m/s, the distance \( d \) can be expressed as:
   \[
   d = 343 \cdot t_A
   \]
   and
   \[
   d = 343 \cdot t_B
   \]
5. Substituting \( t_B = t_A + 5 \) into the second equation:
   \[
   d = 343 \cdot (t_A + 5)
   \]
6. Equating the two expressions for \( d \):
   \[
   343 \cdot t_A = 343 \cdot (t_A + 5)
   \]
7. Simplifying, we find:
   \[
   t_A = t_A + 5
   \]
   This is a contradiction, so we need to re-evaluate the problem setup. The correct approach is to consider the relative timing of the yells and the distances.

8. If Bananastasia yells first and Anastasia yells when she hears Bananastasia, then the time it takes for Bananastasia's yell to reach Anastasia is:
   \[
   t_{BA} = \frac{d}{343}
   \]
9. Anastasia yells immediately upon hearing Bananastasia, so the total time for Anastasia's yell to reach you is:
   \[
   t_A = t_{BA} + \frac{d}{343} = \frac{d}{343} + \frac{d}{343} = \frac{2d}{343}
   \]
10. Given that you hear Anastasia’s yell 5 seconds after Bananastasia’s yell:
    \[
    t_A = t_B + 5
    \]
11. Substituting \( t_B = \frac{d}{343} \) and \( t_A = \frac{2d}{343} \):
    \[
    \frac{2d}{343} = \frac{d}{343} + 5
    \]
12. Solving for \( d \):
    \[
    \frac{2d}{343} - \frac{d}{343} = 5
    \]
    \[
    \frac{d}{343} = 5
    \]
    \[
    d = 5 \times 343
    \]
    \[
    d = 1715 \text{ meters}
    \]

The final answer is \(\boxed{1715}\) meters.