Home > Algebra, Olympiad, Problem Solving > IMC 2014 Day 1 Problem 3

IMC 2014 Day 1 Problem 3


Let {n} be a positive integer. Show that there are positive real numbers {a_0,a_1,...,a_n} such that for each choice of signs the polynomial

\displaystyle \pm a_nx^n\pm a_{n-1}x^{n-1}\pm ... \pm a_1x\pm a_0

has {n} distinct real roots.

IMC 2014 Day 1 Problem 3

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  1. Artur
    August 3, 2014 at 12:25 pm

    Use induction. For the transition step, add a very small free coefficient at the end of the polynomial, so that the old roots don’t change by much.

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