No of ordered pairs satisfying this equation

[cr7einstein]

Homework Statement

We are required to find the no. of ordered pairs $(x,y)$ satisfying the equation

$13+12[tan^{-1}x]=24[ln x]+8[e^x]+6[cos^{-1}y]$. ($[.]$ is the greatest integer function, e.g. $[2.3]=2$, $[5.6]=5$, $[-2.5]=-3$ etc)

Homework Equations

The Attempt at a Solution

The answer happens to be zero. I tried to arrange the terms so that I can show that the ranges on either side of the equation don't overlap, but the logarithmic and exponential terms always make the range the set of real numbers, so that doesn't work. Also, the constraint on the domain is that $x$ must be positive because of the logarithm term. I then tried to study two cases $x>1$ and then $x$ between zero and one. But I haven't made any progress. Any help would be appreciated; thanks in advance!

 

[haruspex]

You are making it much too complicated. Ignore what's inside the [] brackets.

 

[cr7einstein]

@haruspex what do you mean? any suggestion ?

 

[haruspex]

@haruspex what do you mean? any suggestion ?

Write out the equation, treating the terms inside the [] brackets as arbitrary unknowns. What does it look like?
Remember that the [] function itself always returns an integer.

 

[cr7einstein]

@haruspex Thanks! One side is an odd integer and the other an even integer; so no solutions...but is there a 'rigorous' way to do this??

 

[haruspex]

@haruspex Thanks! One side is an odd integer and the other an even integer; so no solutions...but is there a 'rigorous' way to do this??

Take mod 2.

 

[cr7einstein]

@haruspex haha...okay I'll stop now :P. I'll mark this as solved now. I was trying to work it out using the properties of [.] and ranges of the functions involved but looks like it doesn't work here.