Geometric progression of terms involving floor function Which positive real number has the property that $x ,\lfloor x\rfloor,$ and $x-\lfloor x\rfloor$ form a geometric progression (in that order)? 

Hint. Let $ x = a + k $, where $ a = \left\lfloor x \right\rfloor $ and $ 0 \le k < 1 $. Then, we want $$ \frac {k}{a} = \frac {a}{a + k} \implies ak + k^2 = a^2, $$so $ak+k^2$ needs to be an integer. 
