Question 6
A6 (IRE) Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
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′ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
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 $n=q2^r$, $r\in\mathbb N$.
A6″ Prove that for each positive integer $n$, there exist polynomials $f(x),g(x)$ having integer coefficients such that
$$f(x)(x+1)^{2^n}+g(x)(x^{2^n}+1)=2.$$
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
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′ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
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 $n=q2^r$, $r\in\mathbb N$.
A6″ Prove that for each positive integer $n$, there exist polynomials $f(x),g(x)$ having integer coefficients such that
$$f(x)(x+1)^{2^n}+g(x)(x^{2^n}+1)=2.$$