<s>[INST] Finding the minimum value of $a^2+b^2+c^2$ 
Let $a$, $b$ and $c$ be $3$ real numbers satisfying $2 \leq ab+bc+ca$. Find the minimum value of $a^2+b^2+c^2$. 

I've been trying to solve this, but I don't really know how to approach this. I thought of $(a+b+c)^2 = a^2+b^2+c^2 + 2(ab+bc+ca)$, but that gives me  $a+b+c$, which is unknown. How can I solve this?
 [/INST] Notice that by Inequality of arithmetic and geometric means we know that:
$$ 
2ab \leq a^2+b^2; 
\\ 
2ac \leq a^2+c^2; 
\\
2bc \leq b^2+c^2; 
$$
so we can conclude that: 
$$ 
2\left(ab+ac+bc\right) 
\leq 
2\left(a^2+b^2+c^2\right)
\Longrightarrow 
\\ 
\ \ 
\left(ab+ac+bc\right) 
\leq 
\ \ 
\left(a^2+b^2+c^2\right)
\Longrightarrow 
\\ 
\ \ \ \ \ 
\ \ \ \ \ 
\ \ \ \ \ 
\ \ \ \  
2
\leq 
\ \ 
\left(a^2+b^2+c^2\right) 
.
$$ 

Note that this in-equlality is sharp for $a=b=c=\sqrt{\dfrac{2}{3}}$;
for which one can see the value of $a^2+b^2+c^2$ is equal to $2$. 
</s>