IMO 1979 LL SWE61

Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two

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IMO 1979 LL SWE61

Origin: SWE

Problem

Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two sequences such that m k=1 ak \geqm k=1 bk for all m \leqn with equality for m = n. Let f be a convex function defined on the real numbers. Prove that n  k=1 f(ak) \leq n  k=1 f(bk).