Inequalities problem (1 Viewer)

[HeroicPandas]

I found this problem a while ago and worked it out but the end step was a bit weird (i'll post it up when i'm done with a some past papers)

The question is found here:

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=299899&start=60
Post #76, problem 43

If you have any ideas, please post them (dont look at the answer some person posted up in the website)

 

[HeroicPandas]

So i used AM-GM inequality and a + b + c = 3 to get



and



Subtracting inequalities is incorrect yeh?

What about EQUATING these two?

 

[Drongoski]

Cannot be right. Say a = 9, b=16 and c=25

Then LHS = 3+4+5 = 12

RHS = 9x16 + 16x25 + 9x25

 

[HeroicPandas]

Cannot be right. Say a = 9, b=16 and c=25

Oh sorry, I didnt make one thing clear: a + b + c = 3

 

[superSAIyan2]

why is ab + bc + ca < 3

 

[HeroicPandas]

why is ab + bc + ca < 3

Provided:

Square both sides,



Expand,



........................................................[1]





Add all them up to yield:



Sub [1] into inequality above

 

[superSAIyan2]

thanks! but didnt u say u used the AM-GM inequality to prove that? or was that to prove the second inequality?
Í

 

[HeroicPandas]

I used Am-GM Inequality to both both of them lol

Oh, for the first one, i DIDNT use AM-GM STRAIGHT OUT

For the second one however, i used it out of the blue