Find the all arranged triplets ,from positive real numbers ( a,b,c)
that: (a)bc=3,a(b)c=4,ab(c)=5,which (x) is the greatest integer less than or equal to x
Label the original equations (1), (2), and (3) in that order.
First, note that (otherwise , etc., contradiction).
Multiplying all three equations, we get .
We can prove that . Assume that . In order to satisfy (1), b must equal 3, otherwise c would have to be less than 1. If b = 3, . Solving for a via (2) and (3), we obtain and , no solution. Similarly, if , we get the same contradiction. If , , no solutions.
Therefore, . It follows that . I'll let you do the casework (I obtained at least two solutions btw).
(There should be {} with 1,2 and 1,2,4, but LaTeX interprets the brackets differently, and idk how to input them otherwise).