# Another problem from BMO

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#1
It is about inequality, which is shown as follows.
Thanks for help.
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#2
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2 years ago
#3
(Original post by ou_litu)
This is the same as proving So expand the LHS. Try to show this is the case.
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#4
I tried to solve it in your way, but after expansion, no progress at all.
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2 years ago
#5
(Original post by ou_litu)
Jensen's inequality will give you this very quickly. If you're not allowed to assume Jensen, then follow the method of proof!
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2 years ago
#6
(Original post by ou_litu)
I tried to solve it in your way, but after expansion, no progress at all.
Alright, it was just an idea. Though looking at the question from a source, I have to ask, is there a previous part to your question where you prove the inequality ??

Because it seems to be useful.
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#7
(Original post by RDKGames)
Alright, it was just an idea. Though looking at the question from a source, I have to ask, is there a previous part to your question where you prove the inequality ??

Because it seems to be useful.
Sorry, still can't see your pount.
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2 years ago
#8
Another way:

Spoiler:
Show

3(a^3+a^3+b^3+b^3+c^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c) by the rearrangement inequality

a^3+b^3+c^3 >= 3abc by AM-GM so 2(a^3+b^3+c^3) >= 3abc

8(a^3+b^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c+abc)

so 9(a^3+b^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c+abc) + a^3+b^3+c^3 = (a+b+c)^3.

Last edited by username2844924; 2 years ago
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2 years ago
#9
(Original post by ou_litu)
Sorry, still can't see your pount.
So is that a yes to my question? If so, notice that from this inequality we have that     If you can move from there.
Last edited by RDKGames; 2 years ago
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#10
Can't imagine it can be solved in such an easy way.
Thanks a lot.
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#11
(Original post by J843126028)
Another way:

Spoiler:
Show

3(a^3+a^3+b^3+b^3+c^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c) by the rearrangement inequality

a^3+b^3+c^3 >= 3abc by AM-GM so 2(a^3+b^3+c^3) >= 3abc

8(a^3+b^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c+abc)

so 9(a^3+b^3+c^3) >= 3(a^2b+b^2c+c^2a+b^2a+c^2b+a^2c+abc) + a^3+b^3+c^3 = (a+b+c)^3.

Thanks.
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