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\sum_{k=1}^{2n+1}(-1)^k [ k/2 ] P(x + k)=0 for infinitely many real x

Source: Austrian - Polish 2000 APMC

May 4, 2020
floor functionSumpolynomialalgebra

Problem Statement

Find all polynomials P(x)P(x) with real coefficients having the following property: There exists a positive integer n such that the equality ∑k=12n+1(−1)k[k2]P(x+k)=0\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0 holds for infinitely many real numbers xx.
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