2020 AMC 12A Problem 25

The number a = \tfrac{p}{q}, where p and q are relatively prime positive integers, has the property that the sum of all real numbers x satisfying\lfloor x \rfloor \cdot \{x\} = a \cdot x^2is 420, where \lfloor x \rfloor denotes the greatest integer less than or equal to x and \{x\} = x - \lfloor x \rfloor denotes the fractional part of x. What is p + q?

\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332

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