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Putnam 2002 B6

Source:

March 12, 2012
Putnamlinear algebramatrixalgebrapolynomialmodular arithmeticbinomial theorem

Problem Statement

Let pp be a prime number. Prove that the determinant of the matrix [xyzxpypzpxp2yp2zp2] \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} is congruent modulo pp to a product of polynomials of the form ax+by+czax+by+cz, where aa, bb, and cc are integers. (We say two integer polynomials are congruent modulo pp if corresponding coefficients are congruent modulo pp.)
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