Problem 1386

[Solution] First, we prove the existence of $k$.
In fact, when $n$ is even, $\left((x+1)^{n}, x^{n}+1\right)=1$. Therefore, there exist rational coefficient polynomials $f^{*}(x), g^{*}(x)$, such that
$$1=f^{*}(x) \cdot(x+1)^{n}+g^{*}(x)\left(x^{n}+1\right) \text {. }$$

Let $k$ be a common multiple of the denominators of all the coefficients of $f^{*}(x)$ and $g^{*}(x)$, and set $f(x)=$ $k f^{*}(x), g(x)=k g^{*}(x)$, then $f(x), g(x)$ are integer coefficient polynomials, and
$$k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right) .$$

Thus, the existence of $k$ is proven.
Next, we find the minimum value of $k$, denoted as $k_{0}$.
Let $n=2^{\alpha} t$, where $t$ is an odd number and $\alpha$ is a non-negative integer. Let $m=2^{\alpha}$. Then, we have
$$x^{n}+1=\left(x^{m}\right)^{t}+1=\left(x^{m}+1\right) \cdot h(x),$$

where $h(x)$ is an integer coefficient polynomial.
Let the $m$ roots of $x^{m}+1=0$ be
$$w_{j}=e^{i \cdot \frac{2 j-1}{m} \pi}, j=1,2, \cdots, m, m=2^{\alpha} .$$

If a positive integer $k$, and integer coefficient polynomials $f(x), g(x)$ satisfy
$$k=f(x) \cdot(x+1)^{n}+g(x)\left(x^{n}+1\right),$$

then
$$k=f\left(w_{j}\right)\left(w_{j}+1\right)^{n}, j=1,2, \cdots, m .$$

Thus, we have
$$k^{m}=\prod_{j=1}^{m} f\left(w_{j}\right) \cdot \prod_{j=1}^{m}\left(w_{j}+1\right)^{n} \cdot$$

Let
$$\begin{array}{l} \sigma_{1}=\omega_{1}+\omega_{2}+\cdots+\omega_{m}, \\ \sigma_{2}=\omega_{1} \omega_{2}+\omega_{1} \omega_{3}+\cdots+\omega_{m-1} \omega_{m}, \\ \cdots \cdots \cdots \\ \sigma_{m}=\omega_{1} \omega_{2} \cdots \omega_{m} . \end{array}$$

By Vieta's formulas, $\sigma_{j}$ is an integer, $j=1,2, \cdots, m$. Since $\prod_{j=1}^{m} f\left(w_{j}\right)$ is a symmetric polynomial in $\omega_{1}, \omega_{2}, \cdots, \omega_{m}$ with integer coefficients, it can be expressed as a polynomial in $\sigma_{1}, \sigma_{2}, \cdots, \sigma_{m}$ with integer coefficients, and thus it is an integer. Also, because
$$\begin{aligned} & \prod_{j=1}^{m}\left(w_{j}+1\right)^{n}=\left[\prod_{j=1}^{m}\left(w_{j}+1\right)\right]^{n} \\ = & \left(1+\sigma_{1}+\sigma_{2}+\cdots+\sigma_{m}\right)^{n} \\ = & 2^{n} . \end{aligned}$$

Therefore, $2^{n} \mid k^{m}$, so $2^{t} \mid k$, and $k \geqslant 2^{t}$.
On the other hand, we set
$$E(x)=(x+1)\left(x^{3}+1\right) \cdots\left(x^{2 m-1}+1\right)=(x+1)^{m} \cdot F(x) \text {. }$$

For a fixed $j \in\{1,2, \cdots, m\}$, consider the set
$$\left\{\omega_{j}, \omega_{j}^{3}, \omega_{j}^{5}, \cdots, \omega_{j}^{2 m-1}\right\},$$

where the elements are all roots of $x^{m}+1=0$ and are distinct, hence it is the solution set of $x^{m}+1=0$. Thus,
$$\begin{aligned} E\left(W_{j}\right) & =\left(1+\omega_{j}\right)\left(1+\omega_{j}^{3}\right) \cdots\left(1+\omega_{j}^{2 m-1}\right) \\ & =\left(1+\omega_{1}\right)\left(1+\omega_{2}\right) \cdots\left(1+\omega_{m}\right) \\ & =2 \end{aligned}$$

i.e., $\omega_{j}$ is a root of $E(x)-2$. Therefore, we can set
$$G(x)\left(x^{m}+1\right)+2=E(x)=(x+1)^{m} F(x),$$

raising both sides to the $t$-th power, we get
$$G^{*}(x) \cdot\left(x^{m}+1\right)+2^{t}=(x+1)^{n} F^{t}(x)$$

where $G^{*}(x)$ is some integer coefficient polynomial.
Since
$$x^{n}+1=\left(x^{m}+1\right) h(x),$$

where $h(x)$ satisfies $h(-1)=1$, we can set
$$c(x) \cdot(x+1)=h(x)-1 \text {, }$$

raising both sides to the $n$-th power, we get
$$C^{n}(x) \cdot(x+1)^{n}=h(x) \cdot d(x)+1$$

where $d(x)$ is some integer coefficient polynomial.
From (1) and (2), we get
$$\begin{aligned} & G^{*}(x) d(x) \cdot\left(x^{n}+1\right) \\ = & G^{*}(x) \cdot\left(x^{m}+1\right) \cdot d(x) \cdot h(x) \\ = & {\left[(x+1)^{n} F^{t}(x)-2^{t}\right] \cdot\left[C^{n}(x)(x+1)^{n}-1\right] } \\ = & (x+1)^{n} \cdot U(x)+2^{t}, \end{aligned}$$

where $U(x)$ is some integer coefficient polynomial.
Therefore, there exist integer coefficient polynomials $f(x), g(x)$, where
$$f(x)=-U(x), g(x)=G^{*}(x) \cdot d(x),$$

such that
$$f(x) \cdot(x+1)^{n}+g(x) \cdot\left(x^{n}+1\right)=2^{t} .$$

In conclusion, the minimum value of $k$ is $k_{0}=2^{t}$, where $n=2^{\alpha} \cdot t, t$ is an odd number, and $\alpha$ is a non-negative integer.