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Minimum and maximum value of a sum

Source: Moldova IMO-BMO TST 2003, day 3, problem 2.

August 16, 2008

Problem Statement

Let a1,a2,...,a2003≥0 a_1,a_2,...,a_{2003}\geq 0, such that a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2 and a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1. Determine the minimum and maximum value of a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2.
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