Putnam 2005, Problem B5 Let $P(x_1,\ldots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\ldots,x_n$, and suppose that [paraphrased] the Laplacian of $P$ vanishes identically and $x_1^2 + \cdots + x_n^2$ divides $P(x_1,\ldots,x_n)$. Show that $P=0$ identically.
That is: Prove that no nonzero harmonic polynomial is divisible by $x_1^2 + \cdots + x_n^2$. Each of the three solutions reported at http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2005s.pdf (one by Yours Truly) grows naturally out of one approach to the development of the theory of spherical harmonics. See also http://mathoverflow.net/questions/67411/signed-factors-of-harmonic-polynomials for a generalization that appeared on this forum a few weeks ago.