Things you need:
False swipe
A lot of ultra balls (and other kinds of pokeballs if you want like net balls, quick balls, dusk balls)
A pokemon that has thunder wave, spore, sleep or paralyzing powder, or an ice move that can freeze the foe (or all of these if you like:P)
Revives and full restores!

Ok...Let's say you have a pokemon you want to catch, for example Mewtwo which is lvl 70 (at Cerulean cave). First of all, you need two or three strong pokemon that cannot faint easily from Mewtwo's psycho cut. The rest of its moves are guard swap, power swap(or something like that)and amnesia. Amnesia is really helpful because it makes it harder to make the pokemon faint(and that's what we need actually:).
When you encounter it, use firstly electric or ice type moves to get it's HP low. If it doesn't become paralyzed or frozen, use the powder attacks. These effects make it easier to catch, that's why you need them.
After its HP has becomes red, send out the pokemon that has the move false swipe(i used a Parasect lvl 40 that also had slash, spore and something else:P). Use it until Mewtwo has only 1 HP left. After that you only need a bit of luck! You will need to use up to 10 or 15 pokeballs the most.
I caught Mewtwo, Ho-oh and Suicune easily using 10 or so pokeballs. Well, good luck everyone and i hope i helped a bit:)

P.S.You won't need this secret-hint call it whatever you want, if you have AR but...Well, for those who don't...You will also need a good amount of money but i hope you don't have such a problem:).