Školjka

%V0 Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)$$ 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 $n,$ i.e. show that $k_0 = 2^q$ where $q$ is the odd integer determined by $n = q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.