# 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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