If we have real numbers $x, y, z \in [1, 2]$ then what is the maximum of
$$\frac{x^2+y^2+z^2}{xy+xz+yz}$$
I tried to use substitution $x=\frac{3+\sin X}{2}$, $y=\frac{3+\sin Y}{2}$ and $z=\frac{3+\sin Z}{2}$. But the expression became too messy. This is an Olympiad problem (I don't know the source) and I am not allowed to use calculus. I hope someone can provide an insight to this problem!
Let $k$ is a maximal value and $f(x,y,z)=x^2+y^2+z^2-k(xy+xz+yz)$.
Since, $f$ is a convex function of $x$, of $y$ and of $z$, we obtain: $$0=\max_{\{x,y,z\}\subset[1,2]}f=\max_{\{x,y,z\}\subset\{1,2\}}f$$, which for $x=y=1$ and $z=2$ gives $k=\frac{6}{5}$ and we are done!