<s>[INST] maximum value of $\sum (a-b)^2$ 
If $a^2+b^2+c^2=5$ and $a,b,c \in \mathbb{R},$ find the maximum
value of $(a-b)^2+(b-c)^2+(c-a)^2$.

My Try: $(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2(ab+bc+ac)$
$$=10-2(ab+bc+ac)$$
Now this implies $ab+bc+ac\le 5$. So
$$(a-b)^2+(b-c)^2+(c-a)^2 = 10-2(ab+bc+ac) \geq 20.$$
Could some help me to find max of $(a-b)^2+(b-c)^2+(c-a)^2$? Thanks.
 [/INST] $$\sum_{cyc}(a-b)^2=3(a^2+b^2+c^2)-(a+b+c)^2\leq15.$$
The equality occurs for $a+b+c=0,$ which says that we got a maximal value.
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