## Putnam 2010 A3

Suppose that the function has continuous partial derivatives and satisfies the equation for some constants . Prove that if there is a constant such that for all , then is identically zero.

*Putnam 2010 A3*

**Solution:** If we have nothing to prove. Suppose at least one of them is not zero, and define . Then it is easy to see that . Since and is bounded, it must be the constant zero function. Hence .

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