All perfect praise be to Allah the Lord of The Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
There is nothing wrong with accepting donations from banks and other institutions to help the poor and cover the cost of prizes awarded in religious competitions. This is because, in principle, the money of the banks is lawful, but it has been used in a forbidden manner, which is usury and thus the lawful was mixed with the unlawful. However, what the bank gives in the form of donation isn`t the unlawful money itself; rather, it is the money in its possession. Ill-gotten money whose owner is anonymous should be gotten rid of through giving it to charity and spending it to help the common good. And Allah The Almighty Knows Best.