The one charity theorem

The normal scheme for donating to charities is to divide money up among several different charities. The following argument shows why this strategy is often wrong. Both the statement and the proof will be extremely informal: One charity theorem: Assume we have a fixed amount of money to divide between $n$ charities. Assume that utility is a smooth function of the performances of the charities, which in turn depend smoothly on the amount of money each receives.