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Putnam 2004 A4

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December 11, 2004
Putnamalgebrapolynomialcollege contests

Problem Statement

Show that for any positive integer nn there is an integer NN such that the product x1x2⋯xnx_1x_2\cdots x_n can be expressed identically in the form x1x2⋯xn=∑i=1Nci(ai1x1+ai2x2+⋯+ainxn)nx_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n where the cic_i are rational numbers and each aija_{ij} is one of the numbers, −1,0,1.-1,0,1.
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