Do we believe psychic and supernatural powers of Buddha?
You are basically saying "I have doubts about the Buddha's supernatural powers. Can someone prove them to me?". Nobody can do this for you unless the Buddha himself comes and show you. But that's not going to happen. So you have few options to appease your mind.


Meditate on the doubt until you realize that it's just another mental state that arises and falls and that holding on to it is useless and does not lead you to freedom from suffering. 
Take it on faith, accepting the fact that you are not capable of knowing all that is possible with your current state of mind. And then do #1.
Do Samatha meditation, follow the instructions given in the Visuddhimagga that lead to attaining magical powers.
Waste your life trying to scientifically prove or disprove whether supernatural powers are real or not.