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Number of positive roots of polynomial with integer coeff

Source: Turkish NMO 1st Round - 2002 - P28

August 7, 2014
algebrapolynomialalgebra proposed

Problem Statement

How many positive roots does polynomial x2002+a2001x2001+a2000x2000+⋯+a1x+a0x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0 have such that a2001=2002a_{2001} = 2002 and ak=−k−1a_k = -k - 1 for 0\leq k \leq 2000?
<spanclass=′latex−bold′>a)</span> 0<spanclass=′latex−bold′>b)</span> 1<spanclass=′latex−bold′>c)</span> 2<spanclass=′latex−bold′>d)</span> 1001<spanclass=′latex−bold′>e)</span> 2002 <span class='latex-bold'>a)</span>\ 0 \qquad<span class='latex-bold'>b)</span>\ 1 \qquad<span class='latex-bold'>c)</span>\ 2 \qquad<span class='latex-bold'>d)</span>\ 1001 \qquad<span class='latex-bold'>e)</span>\ 2002
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