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Composition of Number Theoretic Functions

Source: India Postals 2015 Set 3

November 7, 2015
number theoryfunction

Problem Statement

Let k∈Nk \in \mathbb{N}, let xkx_k denote the nearest integer to k\sqrt k. Show that for each m∈Nm \in \mathbb {N}, ∑k=1m1xk=f(m)+mf(m)+1\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}, where f(m)f(m) is the integer part of 4m−3−12\frac{\sqrt{4m-3}-1}{2}
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