# Not a Typo

Algebra Level 5

Let $$a,b$$ and $$c$$ be real numbers satisfying $$abc =a+b+c$$.

The maximum value of $$\dfrac1{\sqrt{1+a^2}} + \dfrac1{\sqrt{1+b^2}} + \dfrac c{\sqrt{1+c^2}}$$ can be expressed as $$\dfrac {p \sqrt q}r$$, where $$p,q,r$$ are all positive integers with $$p,r$$ coprime and $$q$$ square-free.

Find $$p+q+r$$.

Clarification: The third term in the addition is indeed $$\dfrac c{\sqrt{1+c^2}}$$, not $$\dfrac 1{\sqrt{1+c^2}}.$$

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