A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.
1. **Definition and Initial Setup**:
   A polynomial \( f(x) \) with integer coefficients is said to be *tri-divisible* if \( 3 \) divides \( f(k) \) for any integer \( k \). We need to determine the necessary and sufficient conditions for a polynomial to be tri-divisible.

2. **Polynomial Representation**:
   Consider a polynomial \( f(x) \) of degree \( n \) with integer coefficients. We can write:
   \[
   f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
   \]
   where \( a_i \in \mathbb{Z} \) for all \( i \).

3. **Tri-Divisibility Condition**:
   For \( f(x) \) to be tri-divisible, \( 3 \) must divide \( f(k) \) for any integer \( k \). This implies that \( f(x) \equiv 0 \pmod{3} \) for all \( x \in \mathbb{Z} \).

4. **Polynomial Modulo 3**:
   Consider the polynomial modulo 3:
   \[
   f(x) \equiv a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \pmod{3}
   \]
   For \( f(x) \equiv 0 \pmod{3} \) for all \( x \in \mathbb{Z} \), each coefficient \( a_i \) must be divisible by 3. Hence, \( a_i \equiv 0 \pmod{3} \) for all \( i \).

5. **Necessary and Sufficient Condition**:
   Therefore, the necessary and sufficient condition for \( f(x) \) to be tri-divisible is that all coefficients of \( f(x) \) are divisible by 3. This can be written as:
   \[
   f(x) = 3g(x)
   \]
   where \( g(x) \) is a polynomial with integer coefficients.

6. **General Form**:
   To express this in a more general form, we can write:
   \[
   f(x) = x(x-1)(x-2)Q(x) + 3(ax^2 + bx + c)
   \]
   where \( Q(x) \in \mathbb{Z}[x] \) and \( a, b, c \in \mathbb{Z} \). This form ensures that \( f(x) \) is tri-divisible because the term \( x(x-1)(x-2) \) is always divisible by 3 for any integer \( x \), and the remaining part \( 3(ax^2 + bx + c) \) is clearly divisible by 3.

Conclusion:
The necessary and sufficient condition for a polynomial \( f(x) \) with integer coefficients to be tri-divisible is that all its coefficients are divisible by 3. This can be expressed as:
\[
\boxed{f(x) = x(x-1)(x-2)Q(x) + 3(ax^2 + bx + c)}
\]
where \( Q(x) \in \mathbb{Z}[x] \) and \( a, b, c \in \mathbb{Z} \).