Question 25
The number $a = \dfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$
\lfloor x \rfloor \cdot \{x\} = a \cdot x^2
$$ is $420,$ 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 $p + q?$
\lfloor x \rfloor \cdot \{x\} = a \cdot x^2
$$ is $420,$ 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 $p + q?$