Find the smallest integer $n$ such that there exist polynomials $f_{1}, f_{2}, \ldots, f_{n}$ such that:

$$
f_{1}(x)^{2}+f_{2}(x)^{2}+\ldots+f_{n}(x)^{2}=x^{2}+7
$$
We show that the equation $a^{2}+b^{2}+c^{2}=7 d^{2}$ has no non-trivial solutions, which concludes the cases for $n=1,2,3$ by looking at the constant coefficient. Moreover, the $f_{i}$ are affine since otherwise the degree would be too high, so there exists an $i$ and a $t$ such that $f_{i}(t)= \pm t$ for $n=4$, reducing it to the case $n=3$ that has been studied. Therefore, $n \geq 5$ and $\left(f_{1}, f_{2}, f_{3}, f_{4}, f_{5}\right)=(1,1,1,2, x)$ work.

Key ideas: When dealing with polynomials, look at the degree, and try to fix particular values when there is a lot of freedom. :