To solve the problem, we need to find the value of \(\frac{f(-1) + f(5)}{f(0) + f(4)}\) given that \(f(x)\) is a polynomial of degree 4 with leading coefficient 1 and satisfies \(f(1) = 1\), \(f(2) = 2\), and \(f(3) = 3\).
### Step 1: Formulate the Polynomial
Since \(f(x)\) is a polynomial of degree 4 with leading coefficient 1, we can express it as:
\[
f(x) = x^4 + ax^3 + bx^2 + cx + d
\]
### Step 2: Use the Given Conditions
We know:
- \(f(1) = 1\)
- \(f(2) = 2\)
- \(f(3) = 3\)
From these conditions, we can derive that:
\[
f(x) - x = 0 \text{ for } x = 1, 2, 3
\]
This means that \(f(x) - x\) has roots at \(x = 1, 2, 3\). Therefore, we can express:
\[
f(x) - x = k(x-1)(x-2)(x-3)(x-r)
\]
where \(r\) is another root and \(k\) is a constant.
### Step 3: Determine the Leading Coefficient
Since the leading coefficient of \(f(x)\) is 1, we have \(k = 1\). Thus:
\[
f(x) - x = (x-1)(x-2)(x-3)(x-r)
\]
This implies:
\[
f(x) = (x-1)(x-2)(x-3)(x-r) + x
\]
### Step 4: Expand the Polynomial
Now, we expand \((x-1)(x-2)(x-3)\):
\[
(x-1)(x-2) = x^2 - 3x + 2
\]
\[
(x^2 - 3x + 2)(x-3) = x^3 - 6x^2 + 11x - 6
\]
Thus:
\[
f(x) = (x^3 - 6x^2 + 11x - 6)(x - r) + x
\]
Expanding this gives:
\[
f(x) = x^4 - rx^3 - 6x^3 + 6rx^2 + 11x^2 - 11rx - 6x + 6r + x
\]
Combining like terms:
\[
f(x) = x^4 + (-r - 6)x^3 + (6r + 11)x^2 + (-11r - 5)x + 6r
\]
### Step 5: Calculate \(f(-1)\), \(f(0)\), \(f(4)\), and \(f(5)\)
1. **Calculate \(f(-1)\)**:
\[
f(-1) = (-1)^4 + (-r - 6)(-1)^3 + (6r + 11)(-1)^2 + (-11r - 5)(-1) + 6r
\]
Simplifying gives:
\[
f(-1) = 1 + (r + 6) + (6r + 11) + (11r + 5) + 6r = 24r + 23
\]
2. **Calculate \(f(0)\)**:
\[
f(0) = 6r
\]
3. **Calculate \(f(4)\)**:
\[
f(4) = 4^4 + (-r - 6)(4^3) + (6r + 11)(4^2) + (-11r - 5)(4) + 6r
\]
Simplifying gives:
\[
f(4) = 256 + (-r - 6)(64) + (6r + 11)(16) + (-11r - 5)(4) + 6r
\]
After simplification, we find \(f(4) = 6(4 - r) + 256\).
4. **Calculate \(f(5)\)**:
\[
f(5) = 5^4 + (-r - 6)(5^3) + (6r + 11)(5^2) + (-11r - 5)(5) + 6r
\]
After simplification, we find \(f(5) = 6(5 - r) + 625\).
### Step 6: Substitute Values into the Expression
Now we substitute the values into the expression:
\[
\frac{f(-1) + f(5)}{f(0) + f(4)} = \frac{(24r + 23) + (6(5 - r) + 625)}{6r + (6(4 - r) + 256)}
\]
### Step 7: Simplify the Expression
After simplifying the numerator and denominator, we will find the final value.
### Final Step: Calculate the Greatest Integer
Finally, we compute the greatest integer value of the simplified expression.
