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Three variables inequality
Headhunter   6
N Apr 30, 2025 by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
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Headhunter
Apr 20, 2025
lbh_qys
Apr 30, 2025
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Headhunter
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Headhunter
#1 h  Apr 20, 2025, 6:58 AM
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$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
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programjames1
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programjames1
#2 h  Apr 20, 2025, 7:20 AM
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Can you give the reference? I think this is from a USAJMO contest in the 2010s.

EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
This post has been edited 2 times. Last edited by programjames1, Apr 20, 2025, 7:23 AM
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Headhunter
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Headhunter
#3 h  Apr 20, 2025, 9:01 AM
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I guess that this problem is from chinese materials at 1990~2004. but I'm not sure. Thanks.
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lbh_qys
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lbh_qys
#4 h  Apr 21, 2025, 3:04 AM
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Hint
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lbh_qys
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lbh_qys
#5 h  Apr 21, 2025, 3:18 AM • 2 Y
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another solution
This post has been edited 1 time. Last edited by lbh_qys, Apr 21, 2025, 3:18 AM
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spy27
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spy27
#6 h  Apr 30, 2025, 5:38 AM
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lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?
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lbh_qys
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lbh_qys
#7 h  Apr 30, 2025, 6:08 AM
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spy27 wrote:
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?

$f(x) = (a+x)^2 + (b+x)^2 + (c+x)^2 $ get minimum at $x=0$ iff $a+b+c=0$.
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