Problem of the Day #50

POTD 2024-01-24

Let p(x) be a polynomial with real coefficients and of degree at most 2023 such that p(x)=\frac{\lfloor x^2 \rfloor}{2} for x=1,2,3 \ldots 2024. If N is the remainder when p(0)+p(2025) is divided by 1000, find N.