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cng99 Mar8-12 11:52 PM

Equation with x,y as exponential functions
 
1. The problem statement, all variables and given/known data

This question was asked in the Indian National Maths Olympiad.

The question is:

Find all the possible real ordered pairs of (x,y) for equation
16^[(x^2) + y] + 16^[x + (y^2)]=1


2. Relevant equations

That was the only equation.

3. The attempt at a solution

Using 0.5 + 0.5 =1, (x^2) + y = -0.25 and (y^2) + x= -0.25
(Because [16^(-0.25)]=0.5 , I'm using decimals here instead of fractions)

Solving them gives one real ordered pair of (-0.5,-0.5). But how do I know if that's the only answer?

cng99 Mar8-12 11:54 PM

Re: Equation with x,y as exponential functions
 
This is my first thread ever, by the way. Tell me if I did something wrong.

epenguin Mar9-12 06:03 PM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by cng99 (Post 3805851)
Solving them gives one real ordered pair of (-0.5,-0.5). But how do I know if that's the only answer?

Have you checked that that is a solution?

epenguin Mar10-12 01:20 AM

Re: Equation with x,y as exponential functions
 
Was up till 3 a.m. here last night for a family arrival, and wrote last post just before going to bed. Had thought to write 'is it plausible that the answer is one or a few pairs of numbers like that?'.

Woke about 7, and in a bit saw what you are supposed to do. It starts looking nice but at the end the algebra got looking ugly.

Then realised you are supposed to describe answer geometrically, not algebraically. :wink:

Have not needed to write anything down.

Wondering whether to go back to bed or have breakfast.

cng99 Mar11-12 12:01 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by epenguin (Post 3807114)
Have you checked that that is a solution?

(-0.5,-0.5) fits in perfectly. But the question is to find all possible real ordered pairs. How do I do that?

cng99 Mar11-12 12:03 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by epenguin (Post 3807551)
Then realised you are supposed to describe answer geometrically, not algebraically. :wink:

What do you mean???

NascentOxygen Mar11-12 12:19 AM

Re: Equation with x,y as exponential functions
 
I tried over two days to see what wolframapha would make of this, but it's not functioning. Anyone know whether the free version has been decommissioned? :grumpy:

cng99 Mar11-12 12:57 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by NascentOxygen (Post 3809074)
I tried over two days to see what wolframapha would make of this, but it's not functioning. Anyone know whether the free version has been decommissioned? :grumpy:

I've tried that already. It would just do a differentiation. The problem can only be solved by a human.

(It might be possible to plot a graph of 16^((x^2) + y) + 16^((y^2) + x) = z, and then intersecting the surface with z=1. But I couldn't find anything that would plot something like that. They all accept y as a function of x. This doesn't really get us anywhere. )

Is there anyway to find the minimum or maximum value of 16^((x^2) + y) + 16^((y^2) + x)?
That might help. Possible ways to do so might be using the Arithmetic Mean >= Geometric mean >= Harmonic mean property. Or maybe by just differentiating it and putting dy/dx=0. But I couldn't find anything there. Do tell me if anyone finds anything.

Mentallic Mar11-12 01:15 AM

Re: Equation with x,y as exponential functions
 
I'm having quite a lot of trouble with this problem, it's really got me stumped. But then again, all the Olympiad questions seem to give me a hard time :redface:

Since we have an exponential of the form [itex]a^b+a^c=1[/itex] and we know that [itex]a^b>0, a^c>0[/itex] then what we must have is that [itex]b<0, c<0[/itex]

So

[tex]x^2+y<0[/tex]
[tex]x+y^2<0[/tex]
And this gives us the inequality
[tex]-1<x<0[/tex]
[tex]-\sqrt{-x}<y<-x^2[/tex]
But from here I can't think of a way to further narrow down our possible solution set.

cng99 Mar11-12 03:20 AM

Re: Equation with x,y as exponential functions
 
Hey, Mentallic. Nice try...

This did help a bit...

http://www.wolframalpha.com/input/?i...B+-1%3C+y+%3C0

cng99 Mar11-12 03:27 AM

Re: Equation with x,y as exponential functions
 
Actually, However, that doesn't narrow down to it.

epenguin Mar11-12 04:27 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by epenguin (Post 3807551)
Then realised you are supposed to describe answer geometrically, not algebraically. :wink:

Quote:

Quote by cng99 (Post 3809059)
What do you mean???

I mean - that I had misread the question :biggrin: (missed the ^) :blushing:

NascentOxygen Mar11-12 07:17 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by cng99 (Post 3809172)

You get some sort of result when you click on this link, do you? As I said previously, wolframalpha seems non-functioning to me. I clicked on your link and for 10 mins watched it spinning endlessly on "Computing ..."

Given this, and in light of my inability to get it to draw even a straight-line graph, I think the owners must have made it regionally selective, so it's ignoring certain IP ranges. When I go to http://www.wolframalpha.com I am presented with a totally blank page. Anyone else?

Maybe this facility comes under the umbrella of "National security ..........?" http://i.imgur.com/L5zX9.gif

cng99 Mar11-12 10:47 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by NascentOxygen (Post 3809366)
Maybe this facility comes under the umbrella of "National security ..........?" http://i.imgur.com/L5zX9.gif


Haha! Probably.

cng99 Mar14-12 04:26 AM

Re: Equation with x,y as exponential functions
 
Hmmmm, no solutions then?

(-0.5,-0.5) only?

epenguin Mar15-12 05:47 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by cng99 (Post 3814513)
Hmmmm, no solutions then?

(-0.5,-0.5) only?

Can hardly be. One equation relating two variables is a continuous curve in general. The present curve must be symmetrical about the 'diagonal' line x = y and go through that point. The curve might have more than one branch but if so no other branch cuts that line. Don't know what else they want you to say about it.

You might therefore try rotating the thing counter clockwise by 45 , then it is symmetrical around the new X axis f(X) = f(-X) and see if that inspires anything.

cng99 Mar15-12 07:42 AM

Re: Equation with x,y as exponential functions
 
Well, if it's a curve, why wouldn't anyone plot it?
Wolfram Alpha plots a curve for 16^[(x^2) + y] + 16^[x + (y^2)]= anything more than 1.

Maybe whatever the curve is, it gets reduced to a point when 16^[(x^2) + y] + 16^[x + (y^2)]= 1

epenguin Mar15-12 09:49 AM

Re: Equation with x,y as exponential functions
 
Quote:

Quote by cng99 (Post 3816462)
Well, if it's a curve, why wouldn't anyone plot it?
Wolfram Alpha plots a curve for 16^[(x^2) + y] + 16^[x + (y^2)]= anything more than 1.

Maybe whatever the curve is, it gets reduced to a point when 16^[(x^2) + y] + 16^[x + (y^2)]= 1

Can you show us what the family looks like, so we see what happens as you reduce this something to 1?


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