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a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0

Source: Polish MO Second Round 1960 p1

August 31, 2024
Sumalgebrainequalities

Problem Statement

Prove that if the real numbers a a and b b are not both equal to zero, then for every natural n n a2n+a2n−1b+a2n−2b2+…+ab2n−1+b2n>0. a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0.
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