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CIIM 2011 Problem 3

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June 9, 2016
CIIM 2011undergraduate

Problem Statement

Let f(x)f(x) be a rational function with complex coefficients whose denominator does not have multiple roots. Let u0,u1,...,unu_0, u_1,... , u_n be the complex roots of ff and w1,w2,...,wmw_1, w_2,..., w_m be the roots of f′f'. Suppose that u0u_0 is a simple root of ff. Prove that ∑k=1m1wk−u0=2∑k=1n1uk−u0. \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.
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