Which of the following is correct ?

Options

A. `[{x}] + {[x]} + [{x^(2) + x + 2}] + {[x^(2) + x + 2]} = 0`
B. `([(200 + 1)/(2)] + [(200 + 2)/(2^(2))]+......oo) + ([10] + [10 + (1)/(10)] + [10+(2)/(10)] + .....[10+(9)/(10)]) = 300`
C. `[[[x]]] = [x]`
D. `[|x|] = |[x]|`

Answer

To solve the problem, we need to evaluate the correctness of the given options based on the properties of the greatest integer function (also known as the floor function) and the fractional part function. Let's analyze each option step by step.

### Step 1: Analyze Option 1

**Option 1:**
\[ \text{greatest integer function of fraction part of } x + \text{fraction part of greatest integer function of } x + \text{greatest integer function of fraction part of } x^2 + x + 2 + \text{fraction part of greatest integer function of } (x^2 + x + 2) = 0 \]

1. **Greatest Integer Function of Fraction Part of \(x\)**:
- The fractional part of \(x\) is \(x - \lfloor x \rfloor\).
- Thus, \(\lfloor \text{fraction part of } x \rfloor = 0\) for any \(x\) that is not an integer.

2. **Fraction Part of Greatest Integer Function of \(x\)**:
- If \(x\) is an integer, \(\lfloor x \rfloor = x\) and the fractional part is \(0\).
- If \(x\) is not an integer, \(\lfloor x \rfloor\) is still an integer, hence the fractional part is \(0\).

3. **Greatest Integer Function of Fraction Part of \(x^2\)**:
- Similar reasoning applies as above; it will also yield \(0\).

4. **Fraction Part of Greatest Integer Function of \(x^2 + x + 2\)**:
- This will also yield \(0\) since \(x^2 + x + 2\) is an integer or has an integer part.

Thus, all components of Option 1 evaluate to \(0\), making the entire expression equal to \(0\). Therefore, **Option 1 is correct**.

### Step 2: Analyze Option 2

**Option 2:**
\[ \text{greatest integer function of } 200 + \frac{1}{2} \text{greatest integer function of } 200 + \frac{2}{2^2} + \ldots + \text{greatest integer function of } 10 + \text{greatest integer function of } 10 + \frac{1}{10} + \ldots + \text{greatest integer function of } 10 + \frac{9}{10} = 300 \]

1. **Evaluate the first series**:
- The first term is \(200\).
- The second term is \(\frac{1}{2} \lfloor 200 \rfloor = 100\).
- The third term is \(\frac{2}{4} \lfloor 200 \rfloor = 50\).
- Continuing this way, the series converges to a sum that is less than \(200\).

2. **Evaluate the second series**:
- Each term \(\lfloor 10 + \frac{k}{10} \rfloor\) for \(k = 0\) to \(9\) will yield \(10\).
- Thus, the sum will be \(10 \times 10 = 100\).

3. **Total Sum**:
- Adding both series gives \(197 + 100 = 297\), which is not equal to \(300\).

Thus, **Option 2 is incorrect**.

### Step 3: Analyze Option 3

**Option 3:**
\[ \text{greatest integer function of greatest integer function of greatest integer function of } x = \text{greatest integer function of } x \]

1. **Evaluate the left side**:
- The greatest integer function of an integer is the integer itself.
- Therefore, the left side simplifies to \(\lfloor \lfloor \lfloor x \rfloor \rfloor \rfloor = \lfloor x \rfloor\).

Thus, **Option 3 is correct**.

### Step 4: Analyze Option 4

**Option 4:**
\[ \text{greatest integer function of } |x| = | \text{greatest integer function of } x | \]

1. **Evaluate with \(x = -3.2\)**:
- \(\lfloor -3.2 \rfloor = -4\) and \(|\lfloor -3.2 \rfloor| = 4\).
- \(|-3.2| = 3.2\) and \(\lfloor 3.2 \rfloor = 3\).

Since \(3 \neq 4\), **Option 4 is incorrect**.

### Conclusion

The correct options are **Option 1 and Option 3**.

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