Relation between problems and algorithms
I think the question is more philosophical than scientific, and indeed, as @raphael mentioned, the problem is in the definition of a "Problem".

Algorithm, in the simplest way, is a function (a mapping). 
It gets an input (instance) $in \in I$ and gives an output $out \in O$. 
For any instance $in$ there is only a single output $out=\mathsf{ALG}(in)$. 
Therefore, the algorithm solves only a single problem — that is, it defines only a single mapping from $I$ to $O$.

True, if you have two problems (mappings), $P_1 : I_1 \to O_1$ and $P_2: I_2 \to O_2$ we can construct a "single" algorithm that solves "both". It gets as an input the pair $(in,type)$. If $type=1$ it returns $P_1(in)$ and otherwise it returns $P_2(in)$.

But again, this fancy algorithm can be seen as solving a single problem from the domain $(I_1\cup I_2) \times \{1,2\}$ into $O_1 \cup O_2$.