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IMO ShortList 2002, algebra problem 2

Source: IMO ShortList 2002, algebra problem 2

September 28, 2004

inequalitiesalgebraSequenceboundedIMO Shortlist

Problem Statement

Let

a_1,a_2,\ldots

be an infinite sequence of real numbers, for which there exists a real number

c

with

0\leq a_i\leq c

for all

i

, such that \left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \text{for all }i,\ j \text{ with } i \neq j. Prove that

c\geq1