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Putnam 2005 B5

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December 5, 2005
Putnamalgebrapolynomialcalculusderivativesystem of equationscollege contests

Problem Statement

Let P(x1,…,xn)P(x_1,\dots,x_n) denote a polynomial with real coefficients in the variables x1,…,xn,x_1,\dots,x_n, and suppose that (a) (∂2∂x12+⋯+∂2∂xn2)P(x1,…,xn)=0\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0 (identically) and that (b) x12+⋯+xn2x_1^2+\cdots+x_n^2 divides P(x1,…,xn).P(x_1,\dots,x_n). Show that P=0P=0 identically.
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