I want to find a smooth map $f:S^n\to S^n$ of degree $k\in \mathbb{Z}$.
If $n=1$ it's easy to find $f$. If $n=2$, let $$g:[-\pi/2, \pi/2]\times [0, 2\pi)\to S^2: (t_1, t_2)\mapsto (\cos t_1\cdot \cos t_2, \cos t_1 \cdot \sin t_2, \sin t_1)$$ be a parametrization of $S^2$, I define the following map $$f(g(t_1, t_2))=g(t_1, kt_2).$$
Is it true that this map is smooth? [I have some doubts about the smothness of this map in the points $(0, 0, \pm 1)$]. If it's not, how can I make it smooth?
Might be somewhat unsatisfying, but the following construction works: choose $k$ distinct points in $S^n$, and take small disks around each of them so that they do not intersect. On each disk, consider a smooth map which "wraps" the disk around the sphere in an orientation-preserving way, sending the center to the north pole and everything near the boundary to the south pole. Now extend this to a smooth map on the whole sphere by sending every thus-undefined point to the south pole.
By construction, the preimage of the north pole contains $k$ points where the function preserves orientation, and hence this function has degree $k$.