These Ceilings Can Hold You (In Frustration)

Algebra Level 5

\[\large f(x)=\sum_{i=1}^{20}\sum_{j=1}^i\left\lceil x+\dfrac{j}{i}\right\rceil\]

Let a function \(f: \mathbb{R}\mapsto \mathbb{Z}\) be defined as described above.

Let \(n\) be a randomly chosen integer. The probability that \(f^{-1}(n)\) exists (i.e there exists a real \(r\) such that \(f(r)=n\)) can be represented as \(\dfrac{p}{q}\) for positive coprime integers \(p,q\). Find \(p+q\).

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