6. Let $A$ be a set of ten real-coefficient quadratic polynomials. 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 quadratic polynomial $f$, the equation
$$
f(n+x)=a x+b
$$

has at most two real roots. Therefore, each polynomial in $A$ appears at most twice in $f_{1}, f_{2}, \cdots, f_{k}$.
Thus, $k \leqslant 20$.
Below is an example for $k=20$.
$$
\text { Let } P_{i}(x)=[x-(2 i-1)](x-2 i)+x \text {, }
$$

where $i=1,2, \cdots, 10$.
Then $f_{2 i-1}=f_{2 i}=P_{i}$.
Hence, $f_{i}(i)=i(1 \leqslant i \leqslant 20)$.