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A sequence of 2000 reals

Source: Lithuanian TST 2005

April 16, 2005

inductionalgebra unsolvedalgebra

Problem Statement

The sequence

a_1, a_2,..., a_{2000}

of real numbers satisfies the condition

a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2

for all

n

1\leq n \leq 2000

. Prove that every element of the sequence is an integer.