MathDB
another incredible polynomial

Source: Moldova TST 2019

March 10, 2019
algebrapolynomial

Problem Statement

Let P(X)=a2n+1X2n+1+a2nX2n+...+a1X+a0P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0 be a polynomial with all positive coefficients. Prove that there exists a permutation (b2n+1,b2n,...,b1,b0)(b_{2n+1},b_{2n},...,b_1,b_0) of numbers (a2n+1,a2n,...,a1,a0)(a_{2n+1},a_{2n},...,a_1,a_0) such that the polynomial Q(X)=b2n+1X2n+1+b2nX2n+...+b1X+b0Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0 has exactly one real root.