Fermat 2004 Q24

---

Problem Statement

The polynomial $f(x)$ satisfies the equation $f(x)-f(x-2)=(2x-1{)}^2$ for all $x$. If $p$ and $q$ are the coefficients of $x^2$ and $x$, respectively, in $f(x)$, then $p+q$ equals

Problem Link

Solution

$$f(x)-f(x-2)=(2x-1{)}^2\mathrm{\forall }x$$ $$x={\displaystyle \frac{1}{2}},⟹f({\displaystyle \frac{1}{2}})=f({\displaystyle \frac{-3}{2}})$$