I started climbing up the hill at 6 am and reached the temple at the top at 6 pm. Next day I started coming down at 6 am and reached the foothill at 6 pm. I walked on the same road.
The road is so short that only one person can walk on it. Although I varied my pace on my way, I never stopped on my way. Then which of the following must be true

Options

A. A) My average speed downhill was greater than that uphill
B. B) At noon, I was at the same spot on both the days
C. C) There must be a point where I reached at the same time on both the days
D. D) There cannot be a spot where I reached at the same time on both the days

Answer

To solve the problem, we need to analyze the information provided about the climbing and descending of the hill. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Journey:**
- The person starts climbing the hill at 6 AM and reaches the temple at 6 PM on the first day. This means the total time taken to climb is 12 hours.
- The next day, the person starts descending at 6 AM and reaches the foothill at 6 PM, also taking 12 hours.

2. **Distance and Speed:**
- Since the person travels the same path both days, the distance covered is the same.
- The average speed for both journeys (uphill and downhill) can be calculated as:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Given that the time taken for both journeys is the same (12 hours), we can conclude that the average speed uphill and downhill must be equal.

3. **Analyzing the Options:**
- **Option A:** "My average speed downhill was greater than the uphill."
- This cannot be true since the average speeds are equal.
- **Option B:** "At noon, I was at the same spot on both days."
- This is also incorrect because at noon on the first day, the person is still climbing, while on the second day, they are descending. They cannot be at the same spot.
- **Option C:** "There must be a point where I reached at the same time on both days."
- This is the correct option. By the Intermediate Value Theorem, since the person is continuously moving and the paths are the same, there must be at least one point on the path where the person is at the same position at the same time on both days.

### Conclusion:
The only statement that must be true is Option C: "There must be a point where I reached at the same time on both days."

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