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prove non-equalities

Source: Romanian National Olympiad 2014, Grade IX, Problem 2

March 2, 2019
algebra

Problem Statement

Let a a be an odd natural that is not a perfect square, and m,n∈N. m,n\in\mathbb{N} . Then
a) {m(a+a)}≠{n(a−a)} \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} b) [m(a+a)]≠[n(a−a)] \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right]
Here, {},[] \{\},[] denotes the fractionary, respectively the integer part.
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