Let f(x) = `ax^(2)+bx+c` where a,b,c are real numbers. If the numbers 2a ,a +b and c are all integers ,then the number of integral values between 1 and 5 that f(x) can takes is______

Answer

To solve the problem, we need to determine how many integral values between 1 and 5 can be taken by the function \( f(x) = ax^2 + bx + c \) given that \( 2a \), \( a + b \), and \( c \) are all integers.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function is given as \( f(x) = ax^2 + bx + c \), where \( a, b, c \) are real numbers. We know that \( 2a \), \( a + b \), and \( c \) are integers.

2. **Setting Up the Conditions**:
Since \( 2a \) is an integer, \( a \) must be a half-integer (i.e., \( a = \frac{k}{2} \) where \( k \) is an integer). This means \( a \) can take values like \( 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots \).

3. **Finding \( b \)**:
From \( a + b \) being an integer, we can express \( b \) as:
\[ b = m - a \]
where \( m \) is an integer. Since \( a \) is a half-integer, \( b \) must also be a half-integer to ensure \( a + b \) is an integer.

4. **Finding \( c \)**:
\( c \) is given as an integer.

5. **Evaluating \( f(x) \) at Integer Points**:
We will evaluate \( f(x) \) at \( x = 1, 2, 3, 4, 5 \):
- \( f(1) = a(1^2) + b(1) + c = a + b + c \)
- \( f(2) = a(2^2) + b(2) + c = 4a + 2b + c \)
- \( f(3) = a(3^2) + b(3) + c = 9a + 3b + c \)
- \( f(4) = a(4^2) + b(4) + c = 16a + 4b + c \)
- \( f(5) = a(5^2) + b(5) + c = 25a + 5b + c \)

6. **Checking Integer Values**:
Since \( a, b, c \) are such that \( 2a \), \( a + b \), and \( c \) are integers, all the expressions above will yield integers because they are linear combinations of integers.

7. **Finding the Range**:
We need to find the values of \( f(x) \) for \( x = 1, 2, 3, 4, 5 \) and check how many of these values fall between 1 and 5.

- For \( f(1) \): \( f(1) = a + b + c \)
- For \( f(2) \): \( f(2) = 4a + 2b + c \)
- For \( f(3) \): \( f(3) = 9a + 3b + c \)
- For \( f(4) \): \( f(4) = 16a + 4b + c \)
- For \( f(5) \): \( f(5) = 25a + 5b + c \)

Given that \( a, b, c \) can take various integer values, we can evaluate these expressions for different combinations of \( a, b, c \) to find how many of these values fall between 1 and 5.

8. **Conclusion**:
By evaluating the function at these points and checking for integer values between 1 and 5, we can conclude the number of integral values that \( f(x) \) can take.

### Final Answer:
The number of integral values between 1 and 5 that \( f(x) \) can take is **5**.