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Polynomial with n roots in (0,1)

Source: Kvant Magazine No. 1 2019 M2544

March 14, 2023
algebrapolynomialinequalitiesKvant

Problem Statement

Let P(x)=xn+a1xn−1+a2xn−2+…+an−1x+anP(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n be a polynomial of degree nn and nn real roots, all of them in the interval (0,1)(0,1). Prove that for all k=1,n‾k=\overline{1,n} the following inequality holds: (−1)k(ak+ak+1+…+an)>0.(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.
Proposed by N. Safaei (Iran)
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