I came across the following question across a math contest and was wondering how to solve it.
Let a be a positive real number that is not an integer and let
$$ n= \left\lfloor \frac {1}{ a- \lfloor a \rfloor } \right\rfloor $$
Prove that $\lfloor (n+1)a \rfloor -1 $ is divisible by $n+1$.
So I played around some values and got that that the quotient would be $\lfloor a \rfloor$. Would it be rigorous enough to prove that $\lfloor a \rfloor (n+1) = \lfloor (n+1)a \rfloor -1 $ if we have the above definition of $n$. Or would you recommend another approach?
Thanks.