What math problems are those algorithms solving?
What value do those solutions have?
Who is collecting those solutions and for what purpose?
Errant: It is an abstract concept and one that's a little hard to understand and something new, which is based on computer computations solving unique "algorithms".
Tigers Eye: What math problems are those algorithms solving?
They are searching for a result of a cryptographic hash function that meets a very restrictive criteria. The only way to find a result that meets the criteria is a brute force search. This requires so many iterations of the computation that it requires a network of computers running in parallel to find a qualifying result within the average 10 minute period the system requires. So, the "problems" the system is "solving" are purely computational and completely benign.
What value do those solutions have?
The result of the cryptographic hash is a digital signature that is computed from all of the transactions preceding it in the accounting ledger. Any change to a transaction or the book keeping data will result in a different digital signature indicating the ledger has been corrupted.
Who is collecting those solutions and for what purpose?
The result of the cryptographic hash is stored in the ledger in a block header. There is a block header generated every 10 minutes on average. Because the result in each block header is computed from transaction data in the current block and data in the preceding block header, each additional block strengthens the security of the transactions further back in the ledger.