The angle A of the `DeltaABC` is obtuse. `x = 2635 - tan B tan C,` if `[x]` denotes the greatest integer function, the value of `[x]` is

Answer

To solve the problem, we need to find the value of \( x = 2635 - \tan B \tan C \) where \( A \) is an obtuse angle in triangle \( ABC \). Here are the steps to find the value of \( [x] \), the greatest integer function of \( x \).

### Step-by-Step Solution:

1. **Understanding the Angles**:
Since angle \( A \) is obtuse, we have:
\[ A > 90^\circ \]
This implies that the sum of angles \( B \) and \( C \) must be less than \( 90^\circ \):
\[ B + C < 90^\circ \]

2. **Identifying the Nature of Angles**:
Since \( B \) and \( C \) are both angles in a triangle and must be positive, we conclude that both \( B \) and \( C \) are acute angles:
\[ 0 < B < 90^\circ \quad \text{and} \quad 0 < C < 90^\circ \]

3. **Using the Tangent Addition Formula**:
We know that:
\[ \tan(B + C) = \frac{\tan B + \tan C}{1 - \tan B \tan C} \]
Since \( B + C < 90^\circ \), we have \( \tan(B + C) > 0 \). This means that the numerator and denominator must both be positive:
- \( \tan B + \tan C > 0 \) (which is always true for acute angles)
- \( 1 - \tan B \tan C > 0 \)

4. **Inequality from the Denominator**:
From \( 1 - \tan B \tan C > 0 \), we can deduce:
\[ \tan B \tan C < 1 \]

5. **Finding Bounds for \( x \)**:
We can now substitute this inequality into our expression for \( x \):
\[ x = 2635 - \tan B \tan C \]
Since \( \tan B \tan C < 1 \), we have:
\[ x > 2635 - 1 = 2634 \]

6. **Upper Bound for \( x \)**:
Since \( \tan B \tan C \) is positive, we also have:
\[ x < 2635 \]

7. **Combining the Inequalities**:
Thus, we can conclude:
\[ 2634 < x < 2635 \]

8. **Applying the Greatest Integer Function**:
The greatest integer function \( [x] \) gives us the largest integer less than or equal to \( x \). Since \( x \) is greater than \( 2634 \) but less than \( 2635 \):
\[ [x] = 2634 \]

### Final Answer:
Therefore, the value of \( [x] \) is:
\[ \boxed{2634} \]