6. Let $A$ be a set of ten real-coefficient polynomials of degree five. It is known that there exist $k$ consecutive positive integers $n+1$, $n+2, \cdots, n+k$, and $f_{i}(x) \in A(1 \leqslant i \leqslant k)$, such that $f_{1}(n+1), f_{2}(n+2), \cdots, f_{k}(n+k)$ form an arithmetic sequence. Find the maximum possible value of $k$.


6. Given that $f_{1}(n+1), f_{2}(n+2), \cdots, f_{k}(n+k)$ form an arithmetic sequence, we know there exist real numbers $a$ and $b$ such that
$$
f_{i}(n+i)=a i+b .
$$

Notice that, for any fifth-degree polynomial $f$, the equation
$$
f(n+x)=a x+b
$$

has at most five real roots. Therefore, each polynomial in $A$ appears at most five times in $f_{1}, f_{2}, \cdots, f_{k}$.
Thus, $k \leqslant 50$.
Below is an example for $k=50$.
$$
\begin{array}{l}
\text { Let } P_{k}(x) \\
=[x-(5 k-4)][x-(5 k-3)] \cdots(x-5 k)+x,
\end{array}
$$

where $k=1,2, \cdots, 10$.
Then $f_{5 k}=f_{5 k-1}=f_{5 k-2}=f_{5 k-3}=f_{5 k-4}=P_{k}$, where $1 \leqslant k \leqslant 10$.
Thus, $f_{k}(k)=k(1 \leqslant k \leqslant 50)$.