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Existence of Roots in Certain Intervals

Source: ILL 1970 - Problem 6.

May 24, 2011
functionalgebra unsolvedalgebra

Problem Statement

There is an equation ∑i=1nbix−ai=c\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c in xx, where all bi>0b_i >0 and {ai}\{a_i\} is a strictly increasing sequence. Prove that it has n−1n-1 roots such that xn−1≤anx_{n-1}\le a_n, and ai≤xia_i \le x_i for each i∈N,1≤i≤n−1i\in\mathbb{N}, 1\le i\le n-1.
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