IMO Shortlist 1996 problem A6

  Avg: 0,0
  Avg: 8,0
Dodao/la: arhiva
2. travnja 2012.
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1996