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[a n^2] is even for infinitely many n

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

July 31, 2011
floor functionalgebrapolynomiallimitnumber theory unsolvednumber theory

Problem Statement

Prove that for any positive real number α\alpha, the number ⌊αn2⌋\lfloor\alpha n^2\rfloor is even for infinitely many positive integers nn.
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