2024 AMC 10B Problem 24

Let

\[ \begin{aligned} P(m) &= \frac{m}{2} + \frac{m^2}{4} \\ &\quad {}+ \frac{m^4}{8} + \frac{m^8}{8}. \end{aligned} \]

How many of the values of P(2022), P(2023), P(2024), and P(2025) are integers?

\textbf{(A)} 0 \qquad \textbf{(B)} 1 \qquad \textbf{(C)} 2 \qquad \textbf{(D)} 3 \qquad \textbf{(E)} 4

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.