Problem: Let $S$ be a compact connected surface which is not homeomorphic to a sphere. Show that there are points on S where the Gauss curvature is positive, negative and zero.
If $S$ is a connected, closed (i.e compact and without boundary) surface, I consider the two cases: $S$ is orientable or $S$ is non-orientable.
From the classification thereom of closed surfaces: If it's orientable, then $\chi(S) = 2-2g$ where $g$ is the number of handles (genus of $S$). Since it's not homeomorphic to a sphere, $\chi(S) \neq 2$ and $g > 0$. If it's non-orientable, then $\chi(S) = 2-h$ where $h$ is the number of crosscaps and since it's not homeomorphic to a sphere, $\chi(S) \neq 2$ and $h > 0$.
So in either case $\chi(S) \leq 0$ and by Gauss-Bonnet $$\int_{S}K = 2\pi \chi(S) \leq 0 $$ the claim follows (elliptic point + intermediate value thereom)
If $S$ is a connected, compact and with boundary, Is this statment even true? In this case, I would have $$\int_{S}K + \int_{\partial S}k_g = 2\pi \chi(S) $$
and I am not entirely sure how this proof would work (if it's true).