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Algebra Difficulty 5.1 AIME, harder Prove it Ukraine

Find all integer nn, that satisfy the following equality:
(n1)(n3)(n5)(n2011)=n(n+2)(n+4)(n+2010). (n-1)(n-3)(n-5)\dots(n-2011) = n(n+2)(n+4)\dots(n+2010).

Solution

If nn is even, then LHS is odd, and RHS is even, and vice versa (for odd nn).

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.