Why is any compact surface with non-trivial boundary homotopy equivalent to bouquet of circles?
It was mentined in "Course homotopy topology" by Fomenko, Fuchs while calculating homotopy groups of compact surfaces, and I have not find any key to this task.
Hint: Every compact surface with boundary can be built as a quotient along the boundary of a closed disk with $k$ open disks removed from its interior (where $k$ is the number of boundary components of the surface).