If `x = (1 ^(2))/(1) =+ (2 ^(2))/( 3) + (3 ^(2))/( 5) +.....+ (1001 ^(2))/( 2001) , y = (1^(2))/(3) + (2 ^(2))/( 5) + (3 ^(2))/(7) + .....+ (1001 ^(2))/(2003), ` then ` ([x -y])/(10)` is equal to
where [.] denotes greatest integer function)

Answer

To solve the problem, we need to find the value of \(\frac{[x - y]}{10}\) where \(x\) and \(y\) are defined as follows:

\[ x = \frac{1^2}{1} + \frac{2^2}{3} + \frac{3^2}{5} + \ldots + \frac{1001^2}{2001} \]

\[ y = \frac{1^2}{3} + \frac{2^2}{5} + \frac{3^2}{7} + \ldots + \frac{1001^2}{2003} \]

### Step 1: Write out the expressions for \(x\) and \(y\)

The expression for \(x\) can be rewritten as:

\[ x = \sum_{n=1}^{1001} \frac{n^2}{2n - 1} \]

And the expression for \(y\) can be rewritten as:

\[ y = \sum_{n=1}^{1001} \frac{n^2}{2n + 1} \]

### Step 2: Find \(x - y\)

Now, we need to find \(x - y\):

\[ x - y = \sum_{n=1}^{1001} \left( \frac{n^2}{2n - 1} - \frac{n^2}{2n + 1} \right) \]

To simplify the expression inside the summation, we can find a common denominator:

\[ \frac{n^2}{2n - 1} - \frac{n^2}{2n + 1} = n^2 \left( \frac{(2n + 1) - (2n - 1)}{(2n - 1)(2n + 1)} \right) = n^2 \left( \frac{2}{(2n - 1)(2n + 1)} \right) \]

Thus,

\[ x - y = \sum_{n=1}^{1001} \frac{2n^2}{(2n - 1)(2n + 1)} \]

### Step 3: Simplify the summation

The term \((2n - 1)(2n + 1)\) can be simplified:

\[ (2n - 1)(2n + 1) = 4n^2 - 1 \]

So we have:

\[ x - y = \sum_{n=1}^{1001} \frac{2n^2}{4n^2 - 1} \]

### Step 4: Factor out constants

We can factor out the constant \(2\):

\[ x - y = 2 \sum_{n=1}^{1001} \frac{n^2}{4n^2 - 1} \]

### Step 5: Evaluate the summation

The term \(\frac{n^2}{4n^2 - 1}\) can be simplified further:

\[ \frac{n^2}{4n^2 - 1} = \frac{n^2}{(2n - 1)(2n + 1)} = \frac{1}{4} + \frac{1}{4(2n - 1)(2n + 1)} \]

Thus, we can express the sum as:

\[ \sum_{n=1}^{1001} \left( \frac{1}{4} + \frac{1}{4(2n - 1)(2n + 1)} \right) \]

### Step 6: Calculate the total

Calculating the first part:

\[ \sum_{n=1}^{1001} \frac{1}{4} = \frac{1001}{4} \]

The second part can be evaluated but is a smaller contribution. For large \(n\), it approaches \(0\).

### Step 7: Final calculation of \(x - y\)

Thus, we can approximate:

\[ x - y \approx 2 \cdot \frac{1001}{4} = \frac{2002}{4} = 500.5 \]

### Step 8: Apply the greatest integer function

Now, applying the greatest integer function:

\[ [x - y] = 500 \]

### Step 9: Divide by 10

Finally, we calculate:

\[ \frac{[x - y]}{10} = \frac{500}{10} = 50 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{50} \]