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sum a_i^3 = (sum a_i)^2 - All-Russian MO 2000 Regional (R4) 11.3

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September 26, 2024
algebranumber theoryInteger

Problem Statement

Sequence of real numbers a1,a2,...,a2000a_1, a_2, . . . , a_{2000} is such that for any natural number nn, 1≤n≤20001\le n \le 2000, the equality a13+a23+...+an3=(a1+a2+...+an)2.a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2. Prove that all terms of this sequence are integers.
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