IMO 1982 LL USA48

Given a finite sequence of complex numbers c1, c2, . . . , cn, show

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IMO 1982 LL USA48

Origin: USA

Problem

Given a finite sequence of complex numbers c1, c2, . . . , cn, show that there exists an integer k (1 \leqk \leqn) such that for every finite sequence a1, a2, . . . , an of real numbers with 1 \geqa1 \geqa2 \geq\cdot \cdot \cdot \geqan \geq0, the following inequality holds: n  m=1 amcmn \leq n  m=1 cm .