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Function involving floors

Source: Central American Olympiad 2006, Problem 3

April 30, 2007
functionfloor functionalgebra proposedalgebra

Problem Statement

For every natural number nn we define f(n)=⌊n+n+12⌋f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor Show that for every integer k≥1k \geq 1 the equation f(f(n))−f(n)=kf(f(n))-f(n)=k has exactly 2k−12k-1 solutions.
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