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Problem 6 of Fourth round - Game strategy with a polynomial

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

September 3, 2019
Game Theorygame strategypolynomialalgebra

Problem Statement

Let f(x)f(x) be a polynomial, such that f(x)=x2015+a1x2014+...+a2014x+a2015f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}. Velly and Polly are taking turns, starting from Velly changing the coefficients aia_i with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?
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