Cyclic Cube Root

Let \(a,b,c\) be integers satisfying the equations

\[a+b+c+1=2014\]

\[\sum_{cyc}\sqrt[3]{a-b+1}=3\]

Find \[\text{min }\text{min}(a,b,c)\]

Details and Assumptions

\(\text{min }\text{min}(a,b,c)\) means the minimum of all possible minimums of \(a,b,c\) where \(a,b,c\) ranges over all possible solutions to the equations.

\(\displaystyle\sum_{cyc}\sqrt[3]{a-b+1}=\sqrt[3]{a-b+1}+\sqrt[3]{b-c+1}+\sqrt[3]{c-a+1}\)

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