Problem of the Month
Problem 4: A Polynomial Sandwich

Let \(a\), \(b\), \(c\), and \(d\) be rational numbers and \(f(x)=ax^3+bx^2+cx+d\). Suppose \(f(n)\) is an integer whenever \(n\) is an integer and that \[\dfrac{1}{3}n^3-n-\dfrac{2}{3}\leq f(n)\leq \dfrac{1}{3}n^3+n^2+2n+\dfrac{4}{3}\] for every integer \(n\) with the possible exception of \(n=-2\).

1. Show that \(a=\dfrac{1}{3}\).
2. Find \(f(10^{2025})-f(10^{2025}-1)\).