<s>[INST] Proving $a = b = c$ under certain conditions For all real a, b, c, prove a = b = c if $$\frac{a^2+b^2+c^2}{3} = (\frac{a+b+c}{3})^2 $$
The first idea that came to mind would be to prove this inequality by contradiction. However, I am unsure about how to go about it. Would using the AM-GM inequality be useful? This is what it could possibly be simplified to make things easier:
$$ 2(a^2 + b^2 + c^2) = 2(ab + ac + bc) $$
Any help would be extremely appreciated.
 [/INST] Write 
$$ 2(a^2 + b^2 + c^2) = 2(ab + ac + bc) $$
As 
$$ (a^2 + b^2 - 2ab) + (c^2 + b^2 - 2cb)+ (a^2 + c^2 - 2ac) = 0$$
Now since $ (a^2 + b^2 - 2ab) = (a-b)^2 $
The expression becomes
$$(a-b)^2 + (a-c)^2 + (c-b)^2 =0 $$ 
Which is only true if $$ a = b = c $$
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