= Solution
The <mean value property for harmonic functions> says that whenever $\overline{B_r(x_0)}\subset\Omega$,
$$
u(x_0)=\frac1{|B_r|}\int_{B_r(x_0)}u
=\frac1{|\partial B_r|}\int_{\partial B_r(x_0)}u.
$$
If $u$ attains its maximum $M$ at an interior point, the average of the nonnegative function $M-u$ over every sufficiently small centred ball is zero. Continuity makes $u=M$ on each such ball, and connectedness propagates this equality throughout the domain. Applying the same argument to $-u$ proves the <Strong maximum principle for harmonic functions>.
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