1996 IMO 第 6 题

A6 (IRE) Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right)$$ for some polynomials $f(x), g(x)$ having integer coefficients. If $k_{0}$ denotes the least such $k$, determine $k_{0}$ as a function of $n$. A6 ${ }^{\prime}$ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right)$$ for some polynomials $f(x), g(x)$ having integer coefficients. If $k_{0}$ denotes the least such $k$, show that $k_{0}=2^{q}$, where $q$ is the odd integer determined by $n=q 2^{r}, r \in \mathbb{N}$. A6" Prove that for each positive integer $n$, there exist polynomials $f(x), g(x)$ having integer coefficients such that $$f(x)(x+1)^{2^{n}}+g(x)\left(x^{2^{n}}+1\right)=2 .$$

A6 (IRE) 令$n$ 为偶正整数。证明对于某些具有整数系数的多项式 $f(x), g(x)$，存在正整数 $k$，使得 $$k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right)$$。如果 $k_{0}$ 表示最小的 $k$，则将 $k_{0}$ 确定为 $n$ 的函数。 A6 ${ }^{\prime}$ 设$n$ 为偶正整数。证明对于某些具有整数系数的多项式 $f(x), g(x)$，存在正整数 $k$，使得 $$k=f(x)(x+1)^{n}+g(x)\left(x^{n}+1\right)$$。如果 $k_{0}$ 表示最小这样的 $k$，则证明 $k_{0}=2^{q}$，其中 $q$ 是由 $n=q 2^{r}, r \in \mathbb{N}$ 确定的奇数整数。 A6" 证明对于每个正整数 $n$，存在具有整数系数的多项式 $f(x), g(x)$，使得 $$f(x)(x+1)^{2^{n}}+g(x)\left(x^{2^{n}}+1\right)=2 。$$