All downloaded files cannot be opened as they are 'damaged and should be moved to the Trash'

Everything I download will not open. OS X says the file is damaged and can't be opened and should be moved to the Trash. Like Adobe Reader, for PDF documents. JPEG files will not open. Office files will not open. Yet the thumbnail preview is perfect and I can flip the pages to see the information is there. I tried to download OpenOffice for the Office documents but it says the same thing, damaged, cannot open, should be moved to the Trash.

^{MacBook Air with Mountain Lion}