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Ukrainian National Mathematical Olympiad, 3rd Round

Algebra Difficulty 5.1 AIME, harder Prove it Ukraine

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

Solution

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

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