Are there any ways to train your brain to work faster? For instance, if you were to practice doing mental arithmetics, seems like all that practice will do for you is memorizing results so you can later take shortcuts when working with them. Which is helpful for that specific purpose, but isn't actually increasing your brain's processing speed.
Is memorizing shortcuts for the specific activity you want to become faster at really all that you can do, or is there something else?
The brain doesn't work like a microprocessor or hardware computer system, it doesn't really have a definable processing speed or any method of increasing it. The brain is a greasy sack of chemicals with some meat-wires throughout and it's immensely less understandable than a computer.
There are drugs that modify brain chemistry, but never permanently and you develop tolerances.
So in short, no.
For increasing your raw intelligence, all of the shit you see claiming to do that is bogus. I've heard of Dual N back tests doing so, but the improvement fades after you quit doing them.
You get smarter/better at problem solving by doing really hard problems a lot. There's not shortcut. There's no game. Honestly, go pick up your text book, look at all the problems you weren't able to solve, and begin solving them.
If you want to become better at mental mathematics, look up Arthur Benjamin's TED talk, and then his book once you're thoroughly convinced he's amazing at it (he is).