On the contrary, I not only proved them, I LINKED to the answers. For instance: Beryllium trigger isotopes can have as little as a 53 day half-life. Polonium 210, a Man-made isotope that can *only* be created in nuclear reactors or cyclotrons, has a 140 day half-life.
Actually polonium is a naturally occurring element, including po-210. It was discovered in 1898 by Mme. Curie. It is rare, but it does occur in nature. It is simply easier to make it from the more abundant bismuth or lead than to try to separate it from the uranium ore, or to try to separate the 25 isotopes from each other. And po-210 is proportionately more rare due to its short half life. The interesting thing about that link is that it describes how to refine polonium using purely physical processes (melting, evaporating) which you claimed were not possible.