2

Part of 1999 Taiwan National Olympiad

Problems(1)

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

Source: 8-th Taiwanese Mathematical Olympiad 1999

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

be a sequence of nonnegative integers such that for any

i,j

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

a_{n}=[nx]\forall n

number theory proposednumber theory