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Find all pairs of positive real numbers

Source: Vietnam TST 1997, Problem 5

July 28, 2008

logarithmsalgebra unsolvedalgebra

Problem Statement

Find all pairs of positive real numbers

(a, b)

such that for every

n \in\mathbb{N}^*

and every real root

x_n

of the equation 4n^2x \equal{} \log_2(2n^2x \plus{} 1) we always have a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n.