For any positive integer , let denote the closest integer to . Evaluate:

*(Putnam 2001)*

**Solution**

We begin by the simple observations that and . Combining this we get that

Hence

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# An arithmotheoretic sum

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For any positive integer , let denote the closest integer to . Evaluate:

*(Putnam 2001)*

**Solution**

We begin by the simple observations that and . Combining this we get that

Hence