Let `f:NrarrN` be a function such `x-f(x)=19[(x)/(19)]-90[(f(x))/(90)],AAx in N` , where [.] denotes the greatest integer function and [.] denotes the greatest integers function and `1900ltf(1990)lt2000`, then possible value of `f(1990)` is

Answer

To solve the problem, we need to analyze the given function and the constraints provided. Let's break down the solution step by step.

### Step 1: Understand the given function
We are given the equation:
\[ x - f(x) = 19\left[\frac{x}{19}\right] - 90\left[\frac{f(x)}{90}\right] \]
where \([.]\) denotes the greatest integer function. We need to find \(f(1990)\) under the condition \(1900 < f(1990) < 2000\).

### Step 2: Analyze the range of \(f(1990)\)
Since \(f(1990)\) must be a natural number between 1900 and 2000, we can express this as:
\[ 1900 < f(1990) < 2000 \]

### Step 3: Divide the range by 90
To use the greatest integer function effectively, we divide the bounds by 90:
\[ \frac{1900}{90} < \frac{f(1990)}{90} < \frac{2000}{90} \]
Calculating these values:
\[ \frac{1900}{90} \approx 21.11 \quad \text{and} \quad \frac{2000}{90} \approx 22.22 \]
This gives us:
\[ 21.11 < \frac{f(1990)}{90} < 22.22 \]
Thus, \(\frac{f(1990)}{90}\) can either be 21 or 22 (as they are the only integers in this range).

### Step 4: Case 1 - \(f(1990) = 21 \times 90\)
If \(\frac{f(1990)}{90} = 21\):
\[ f(1990) = 21 \times 90 = 1890 \]
However, this value does not satisfy \(f(1990) > 1900\).

### Step 5: Case 2 - \(f(1990) = 22 \times 90\)
If \(\frac{f(1990)}{90} = 22\):
\[ f(1990) = 22 \times 90 = 1980 \]
This value satisfies \(f(1990) < 2000\) and \(f(1990) > 1900\).

### Step 6: Check for other possible values
Since we have established that \(f(1990)\) could also be 21 or 22, we can check:
- If \(f(1990) = 21 \times 90 + k\) for \(k = 0, 1, \ldots, 89\), we find that \(f(1990)\) must still be less than 2000.
- If \(f(1990) = 22 \times 90 + k\) for \(k = 0, 1, \ldots, 89\), we find that \(f(1990)\) must still be less than 2000.

### Conclusion
The only valid solution for \(f(1990)\) that satisfies all conditions is:
\[ f(1990) = 1980 \]

### Final Answer
Thus, the possible value of \(f(1990)\) is:
\[ \boxed{1980} \]