Let `r,s` and `t` be the roots of the equation `8x^(3)+1001x+2008=0`. If `99lamda=(r+s)^(3)+(s+t)^(3)+(t+r)^(3)`, the value of `[lamda]` (where [.] denotes the greatest integer function) is ____

Explanation

To solve the problem step by step, we will analyze the given polynomial equation and apply Vieta's formulas to find the required value of \(\lambda\).

### Step 1: Identify the roots and apply Vieta's formulas
The given polynomial is:
\[ 8x^3 + 1001x + 2008 = 0 \]
Let the roots be \(r\), \(s\), and \(t\). According to Vieta's formulas:
- The sum of the roots \(r + s + t = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} = 0\)
- The sum of the products of the roots taken two at a time \(rs + rt + st = \frac{\text{coefficient of } x}{\text{coefficient of } x^3} = \frac{1001}{8}\)
- The product of the roots \(rst = -\frac{\text{constant term}}{\text{coefficient of } x^3} = -\frac{2008}{8} = -251\)

### Step 2: Define new variables for sums of pairs of roots
Let:
\[ a = r + s, \quad b = s + t, \quad c = t + r \]
From the first step, we know:
\[ a + b + c = (r + s) + (s + t) + (t + r) = 2(r + s + t) = 0 \]

### Step 3: Use the identity for the sum of cubes
We use the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Since \(a + b + c = 0\), we have:
\[ a^3 + b^3 + c^3 = 3abc \]

### Step 4: Calculate \(abc\)
We know:
\[ abc = (r+s)(s+t)(t+r) \]
Expanding this:
\[ abc = (r+s)(s+t)(t+r) = (r+s+t)(rs + rt + st) - rst \]
Since \(r+s+t = 0\), this simplifies to:
\[ abc = -rst = 251 \]

### Step 5: Calculate \(a^3 + b^3 + c^3\)
Thus:
\[ a^3 + b^3 + c^3 = 3abc = 3 \times 251 = 753 \]

### Step 6: Relate to \(\lambda\)
We are given:
\[ 99\lambda = a^3 + b^3 + c^3 = 753 \]
Therefore:
\[ \lambda = \frac{753}{99} \]

### Step 7: Simplify \(\lambda\)
Calculating \(\frac{753}{99}\):
\[ \lambda = 7.60606060606\ldots \]

### Step 8: Apply the greatest integer function
The greatest integer function \([\lambda]\) gives us:
\[ [\lambda] = 7 \]

### Final Answer
The value of \([\lambda]\) is:
\[ \boxed{7} \]