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Calculus Level 5

How many real numbers \(I\) are there such that \[I=\oint_{C}\frac{y}{x^2+y^2}dx-\frac{x}{x^2+y^2}dy\]

for some simple closed curve \(C\) that does not run through the origin?

If you come to the conclusion that there are infinitely many such \(I\), enter 666.

Special Case

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