MathDB
k = f(x)*(x+1)^n + g(x)*(x^n + 1)

Source: IMO Shortlist 1996, A6

August 9, 2008
algebrapolynomialfunctionIMO Shortlist

Problem Statement

Let n n be an even positive integer. Prove that there exists a positive inter k k such that k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1) for some polynomials f(x),g(x) f(x), g(x) having integer coefficients. If k0 k_0 denotes the least such k, k, determine k0 k_0 as a function of n, n, i.e. show that k_0 \equal{} 2^q where q q is the odd integer determined by n \equal{} q \cdot 2^r, r \in \mathbb{N}. Note: This is variant A6' of the three variants given for this problem.
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