## zmudz one year ago Prove that if $$x,y,z$$ are positive real numbers then the following inequality holds $$\frac {x+y}{x^2+y^2} + \frac {y+z}{y^2+z^2} + \frac {z+x}{z^2+x^2} \leq \frac 1x + \frac 1y + \frac 1z .$$

1. jtvatsim

Have you attempted anything already? Just so I don't go down a dead end path if you've already tried it. :)

2. zmudz

no, I am stuck :/

3. jtvatsim

k, what class is this for so I know what level of math to use?

4. zmudz

precalculus

5. jtvatsim

alright.

6. zmudz

thank you!

7. jtvatsim

Right now, I am going to try to get the fractions to have the same denominators, the algebra is very messy for now... but I'm going to see where it takes me. I'll do this on scrap paper, rather than on the computer. But know that I am trying to solve it. ;)

8. zmudz

sounds good, I appreciate this a lot!

9. jtvatsim

hmm... this one is nasty... I'm trying a new approach... :)

10. jtvatsim

Almost have something...

11. jtvatsim

Ahh... so close. How have your classes been proving things in precalculus?

12. jtvatsim

Because I could make an argument using around 6-8 cases, but it's a bit tedious... :)

13. jtvatsim

Sorry, I'm struggling a bit. I might just type up a pdf to show the solution. It's a bit messy to look at, but I can't see a simpler method yet.

14. zmudz

AM/GM inequality, maxima minima, induction, mean inequality chain

15. zmudz

also cauchy schwartz

16. zmudz

take your time, thanks. pdf and any help at all is great :)

17. anonymous

sorry to get off topic, how do you type out math latex in a pdf?

18. jtvatsim

I use overleaf.com it's web-based and allows you to download a pdf after typing LaTeX.

19. anonymous

thankyou

20. jtvatsim

Here's my attempt, I didn't finish the proof, because my method was brute force and very tiring...

21. IrishBoy123

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