It depends on how you define "Programming Problem".
In real-world projects, the answer is definitely a clear NO. Most of the problems are not even technical problems, but communication problems, requirements that are unclear, etc.
Then you have whole subjects of problem-classes that require next to no algorithms. For example, GUIs are often straightforward to "program", but the actual problem involved is to have a good design (from a useability point of view, not just the graphical appearance).
There are some fields, where the problems tend to be much more algorithmic by the nature of that field though. For example, AI is a prime subject, where algorithms are at the core. Graphics can be algorithm intensive, but it depends on what exactly is meant with "Graphics Programming".
In general, if the problem you are solving programmatically is suitable for a mathematical representation, then you are entering the algorithmic area. Of course, this is just a rough indicator, as you may create mathematical models for pretty much everything. But for most things you wouldn't normally consider doing so.
Final example: If the problem is to create a GUI that allows entering data for business objects, you wouldn't think about mathematical formulations much. If, however, the problem is to create a GUI that is dynamically changing and relocates elements based on some importance value, you are much more likely to end up with a mathematical model and an algorithmic implementation.