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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