MathDB

Given a polynomial

P\in \mathbb{Z}[X]

of degree

k

, show that there always exist

2d

distinct integers

x_1, x_2, \cdots x_{2d}

such that

P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})

for some

d\le k+1

.
[Extra: Is this still true if

d\le k

? (Of course false for linear polynomials, but what about higher degree?)]

Problem Statement