What is the reason for my teeth hurting every single time I bite into something cold?

Sensitivity. Try a tooth paste, such as sensodyne, this should help.
Couple options. You may have a cracked tooth, you may just have a hypersensitive tooth, but more likely you may have decay or a leaking filling that is creating the increased sense to cold. Especially if it is an isolated tooth that is starting to be sense, you should go see you dentist to have it checked out.