# No of ordered pairs satisfying this equation

• cr7einstein
In summary, the conversation discusses finding the number of ordered pairs (x,y) that satisfy the equation 13+12[tan^-1(x)]=24[ln(x)]+8[e^x]+6[cos^-1(y)], where [.] represents the greatest integer function. The solution is found to be zero through a simplified approach and using the properties of the greatest integer function.

## 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)

## 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!

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

@haruspex what do you mean? any suggestion ?

cr7einstein said:
@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??

Last edited:
Delta2
cr7einstein said:
@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.

@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.