<s>[INST] Beautiful evaluation inequality a,b,c>0 are such that $a^2+b^2+c^2=4$ and $4(a^2+2)=(a^2+b+c)^2$. What is the biggest possible value for a+b+c?
I tried a lot of stuff like $a^2=4-b^2-c^2$. And i think it's somehow connected to QM$\geq$AM where AM is (a+b+c)/3. 
 [/INST] Using the first relation, the left hand side of second relation can be written as $$4(a^2+2)=(a^2+b^2+c^2)(a^2+1^2+1^2)$$ On the right hand side we can apply the Cauchy-Schwarz inequality, so that $$4(a^2+2)=(a^2+b^2+c^2)(a^2+1^2+1^2)\overset{C.S.}\ge(a^2+b+c)^2$$ But due to the second relation we know that it holds actually with equality. Hence $(a,b,c)$ and $(a,1,1)$ must be linearly dependent. Since $a>0$ the only possible solution is $$(a,b,c)=1\cdot(a,1,1)$$ Thus $b=1$, $c=1$ and by substitution $a=\sqrt{2}$, which gives that $$a+b+c=\sqrt{2}+2$$
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