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$a_i+a_j\leq a_{i+j}\leq a_i+a_j+1$

Source: 8-th Taiwanese Mathematical Olympiad 1999

January 20, 2007

number theory proposednumber theory

Problem Statement

Let

a_{1},a_{2},...,a_{1999}

be a sequence of nonnegative integers such that for any

i,j

with

i+j\leq 1999

a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1

. Prove that there exists a real number

x

such that

a_{n}=[nx]\forall n