$\Omega$ simply connected region and $u$ is continuous on $\overline\Omega$. $u$ fulfill the mean value propery for every ball $B(x,r)$ s.t. $B(x,r)\subset \Omega$
$u(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}u(y)\ d\sigma(y)$
Prove u is harmonic
Hint: Assume $\Delta u \neq0$ in some ball $B(a,R)\subset \Omega $ consider auxiliary function $v$ wich solves the boundary condition
$-\Delta v=0 \ $in $B(a,R)$
$v=0 $on $\partial B(a,R)$ and use it appropriately
I don't know how to use the Hint, Can someone help me please