XLVII OM - III - Problem 1

Determine all pairs $ (n,r) $, where $ n $ is a positive integer and $ r $ is a real number, for which the polynomial $ (x + 1)^n - r $ is divisible by the polynomial $ 2x^2 + 2x + 1 $.
Let $ Q_n(x) $ and $ R_n(x) $ denote the quotient and remainder, respectively, when the polynomial $ (x + 1)^n $ is divided by $ 2x^2 + 2x + 1 $. The pair $ (n,r) $ is one of the sought pairs if and only if $ R_n(x) $ is a constant polynomial, identically equal to $ r $.
For $ n = 1,2,3,4 $ we have:






Therefore,






For every integer $ n \geq 0 $, the following equality holds:






Hence, $ R_{n+4}(x) = -\frac{1}{4}R_n(x) $. From this and the formulas (1), we obtain by induction the following equalities (for $ k = 1,2,3,\ldots $):






As we can see, $ R_n(x) $ is a constant polynomial only for $ n = 4k $; its constant value is then $ ( -\frac{1}{4})^k $.
Conclusion: the sought pairs $ (n,r) $ have the form $ (4k,(-\frac{1}{4})^k) $, where $ k $ is a positive integer.