MathDB

Find the greatest real number...

Source: Vietnam TST 1997, Problem 3

July 28, 2008

inductioninequalitiesalgebrapolynomialnumber theory unsolvednumber theory

Problem Statement

Find the greatest real number

\alpha

for which there exists a sequence of infinitive integers

(a_n)

, ( n \equal{} 1, 2, 3, \ldots) satisfying the following conditions: 1)

a_n > 1997n

for every

n \in\mathbb{N}^{*}

; 2) For every

n\ge 2

U_n\ge a^{\alpha}_n

, where U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}.