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This problem asks for an algorithm to determine when and where two trains traveling from Madrid and Barcelona will meet.

Let's break it down step by step:

### Given:
1. Train 1 leaves Madrid at 12:00, traveling towards Barcelona at **200 km/h**.
2. Train 2 leaves Barcelona at **12:30**, traveling towards Madrid at **250 km/h**.
3. The distance between Madrid and Barcelona is **550 km**.

### Step-by-Step Solution:
1. **Train 1’s travel before Train 2 starts:**
   - Train 1 travels for 30 minutes (0.5 hours) before Train 2 starts.
   - In this time, Train 1 covers:
     \[
     \text{Distance} = 200 \, \text{km/h} \times 0.5 \, \text{hours} = 100 \, \text{km}.
     \]
   So, after 30 minutes, Train 1 is **100 km** closer to Barcelona. The remaining distance between the two trains is:
   \[
   550 \, \text{km} - 100 \, \text{km} = 450 \, \text{km}.
   \]

2. **Relative speed of both trains:**
   - Once Train 2 starts, the two trains are moving towards each other.
   - The relative speed of the two trains is:
     \[
     200 \, \text{km/h} + 250 \, \text{km/h} = 450 \, \text{km/h}.
     \]

3. **Time to meet after Train 2 starts:**
   - To cover the remaining **450 km** at a combined speed of **450 km/h**, the time required is:
     \[
     \text{Time} = \frac{450 \, \text{km}}{450 \, \text{km/h}} = 1 \, \text{hour}.
     \]

4. **Final meeting time:**
   - Train 2 starts at 12:30, and they meet 1 hour later, at:
     \[
     12:30 + 1 \, \text{hour} = 13:30.
     \]

5. **Location where they meet:**
   - In 1 hour, Train 1 will travel:
     \[
     200 \, \text{km/h} \times 1 \, \text{hour} = 200 \, \text{km}.
     \]
   So, Train 1 covers an additional **200 km** after Train 2 starts, which means they meet **200 km** from the point Train 1 was at 12:30. Since Train 1 had already traveled **100 km** by 12:30, they meet **300 km** from Madrid.

### Conclusion:
- **Time of meeting:** 13:30
- **Location of meeting:** 300 km from Madrid, or equivalently, 250 km from Barcelona.

Would you like more details or clarification on any step?

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Here are 5 related questions:
1. How would the solution change if both trains traveled at the same speed?
2. What if Train 2 started 1 hour later instead of 30 minutes?
3. How can this problem be solved graphically?
4. How would adding a stop for one of the trains affect the meeting point and time?
5. How would the algorithm adapt to different speeds for different segments of the journey?

**Tip:** In relative speed problems, the key is combining the speeds of objects moving towards or away from each other.

Diseñe un algoritmo que determine cuándo y dónde se cruzarán ambos trenes. Un tren sale de Madrid a las 12:00 y viaja a 200 km/h en dirección Barcelona. Media hora más tarde, sale un tren de Barcelona en dirección contraria a 250 km/h. La distancia entre ambas ciudades es de 550 km.

Solve the problem of determining when and where two trains traveling from Madrid and Barcelona will meet. One train departs at 12:00 traveling at 200 km/h, and the other starts 30 minutes later at 250 km/h. The total distance between the cities is 550 km.

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