That unanswered prayers prove that God does not exist?

No, if God does not answer a prayer then you were not meant to receive what ever you asked for. God gives us everything we have: Health, a home, food, money, every thing we have God has given to us. If we do not get what we have asked for (A friend to be in better health, more money etc.) that means we were not meant to have it.

Another view:

Prayer serves a basic human need; the need to do something when there is nothing that can be done. It is impossible to prove God exists, and even if there is a God, he may not be interested in answering prayers. Scientific studies at Harvard, Duke and elsewhere provide evidence that praying for a sick person makes no difference in the health or recovery of that person. However, there is no harm in praying for a sick person, and it might help the person who is praying to cope with a crisis affecting a loved one, and it might make the sick person feel better to know that someone cares enough to pray for them.

Third answer:

i think that if they dont come true it doesn't mean that God isn't real, it just means he has something more important before yours