IMO 1974 LL SWE32

Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb

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IMO 1974 LL SWE32

Origin: SWE

Problem

Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb for k = 1, 2, . . ., n. If m1 = 1 n(a1 + a2 + \cdot \cdot \cdot + an) and m2 = 1 n(a2 1 + a2 2 + \cdot \cdot \cdot + a2 n), prove that m2 \leq(a+b)2 4ab m2 1 and find a necessary and sufficient condition for equality.