Zi Song's Gargantuan Root

Algebra Level 5

The sum of all real numbers \(x\) such that

\[x = \frac{1}{29}(3 - \sqrt[2013]{29x - 1})\]

can be written as \(\frac{a\sqrt{b}}{c}\), where \(a\) and \(c\) are positive coprime integers and \(b\) is a positive integer that is not divisible by the square of a prime. Find \(a + b + c\).

This problem is posed by Zi Song Y.

Details and assumptions

Note: \(a, b\) and \(c\) could be 1. For example, if the sum is \( 1 = \frac{1 \sqrt{1} } {1} \), then your answer is \(a+b+c = 3 \).


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