Do we believe psychic and supernatural powers of Buddha? I have read from scriptures about psychic and supernatural powers of Buddha and arahats: for example, just after birth the Buddha walked some steps on lotus flowers, elevated during meditation in his childhood, occurrence of earthquake when he touched the earth at the moment of enlightenment, ability to spread water and fire, communication with gods without physical body, ability to read mind of others, recall past lives etc.

The above are accepted only based on our faith to Buddha is reasonable: but in front of rationalists' point of view the above can't be explained; and so,


Do we have to maintain silence as Buddhists?
Do supernatural and psychic phenomena really exist, if so why does nobody take the challenge of rationalists (see e.g. List of prizes for evidence of the paranormal)


I mean not "psychic powers" up to Buddha's level; just for example to describe precisely certain object in a black box (also refer to Abraham Kovoor's challenge).
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.