Question

Prove that if \(x,y,z\) are positive real numbers then the following inequality holds
\( \frac {x+y}{x^2+y^2} + \frac {y+z}{y^2+z^2} + \frac {z+x}{z^2+x^2} \leq \frac 1x + \frac 1y + \frac 1z . \)

Answer 1

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

Answer 2

no, I am stuck :/

Answer 3

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

Answer 4

precalculus

Answer 5

alright.

Answer 6

thank you!

Answer 7

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. ;)

Answer 8

sounds good, I appreciate this a lot!

Answer 9

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

Answer 10

Almost have something...

Answer 11

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

Answer 12

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

Answer 13

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.

Answer 14

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

Answer 15

also cauchy schwartz

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

thankyou

Answer 20

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

Answer 21

.