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2006 SMT Team Round #11 - Determine Sum of Roots

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August 22, 2011
algebrapolynomial

Problem Statement

Polynomial P(x)=c2006x2006+c2005x2005+…+c1x+c0P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0 has roots r1,r2,…,r2006r_1,r_2,\ldots,r_{2006}. The coefficients satisfy 2icic2006−i=2jcjc2006−j2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j} for all pairs of integers 0≤i,j≤20060\le i,j\le2006. Given that ∑i≠j,i=1,j=12006rirj=42\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42, determine ∑i=12006(r1+r2+…+r2006)\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006}).
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