No, they are not cheating. The reason they are doing so much damage is that they have upgraded their in-game items to do so. If you have a high armor rating and they are still doing lots of damage, try buying some equipment with magic resist attributes.
It is also very important to make sure enemy champions are not "fed" early in the game. "Feeding" a champion is the act of allowing them to accrue a lot of kills, rapidly increasing their level and gold and allowing them to deal comparatively massive damage.