Hi! I need help with this inequality.
I have to prove that for any real numbers $a,b,c$ such that $a^2 + b^2 + c^2 =1$ the following inequality holds
$$2\sqrt{2} \leq |a+b+c| + |a+b-c| + |a+c-b| + |b+c-a|$$
#Proving an inequality
steady root
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snaky_man
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oblique echo
Hi! I need help with this inequality.
I have to prove that for any real numbers...
So at some point you probably want to introduce a^2 + b^2 + c^2.
steady root
Yes, I am aware, the problem is, I haven't found a good way to do it
$|x| = \sqrt{x^2}$, then you can use the condition in terms like $(a+b+c)^2$, but it didn't get me anywhere
latent pierBOT
snaky_man
steady root
$a^2 + b^2 + c^2 = 1 \Rightarrow a^2,b^2,c^2 \in [0,1]$ this means that $a,b,c \in [-1,1]$ which allows some trigonometric substitution but this also doesn't seem to great
latent pierBOT
snaky_man
oblique echo
Okay, feels like you're barking up the wrong tree with all this.
I'm gonna tell you what I see.
What's |a + b + c|^2?
steady root
(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca)
oblique echo
There's your 1.
steady root
I know
steady root
$|x| = \sqrt{x^2}$, then you can use the condition in terms like $(a+b+c)^2$, bu...
that's what I meant here
steady root
Why do you have the square root?
using $|a+b+c| = \sqrt{(a+b+c)^2}$ is the easiest way to get to the $(a+b+c)^2$ term I guess
latent pierBOT
snaky_man
oblique echo
using $|a+b+c| = \sqrt{(a+b+c)^2}$ is the easiest way to get to the $(a+b+c)^2$ ...
Disagree.
Also, that's not exactly why you have it.
At least, that's not the answer I was looking for.
steady root
wait, do you want me to explain why is there even a square root in the first place?
oblique echo
...I think so? I'm not quite sure what you mean, but do that and we'll see.
steady root
yes
oblique echo
What else is equal to |a + b + c| that has (a + b + c)^2 in it?
oblique echo
Right.
latent pierBOT
snaky_man
oblique echo
That's correct.
steady root
hmm
steady root
now it looks kind of like some form of cauchy-schwarz inequality
other than that I don't see how this helps
oblique echo
You know things like "Cauchy-Schwarz inequality" but you don't see how this helps?
steady root
yes?
steady root
is it really easy or what?
oblique echo
I mean, I just don't see how you can dismiss it out of hand. Like, it's not a magic bullet, but it's a start of a chain of logic.
steady root
okay, I will think about it
steady root
I still have no clue on how to proceed, can you give me a hint?
steady root
bruh got left on read
oblique echo
I still have no clue on how to proceed, can you give me a hint?
...start with |a + b + c| + |a + b - c| + |a + c - b| + |b + c - a|. Multiply by |a + b + c|. Divide by |a + b + c|.
steady root
To be honest, I don't know, I give up
steady root
Sad
Don't give up soldier
The fight starts once you feel like giving up
@steady root
...
I spent so much time on it and got nothing
I'm tired
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steady root
+close