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Finding natural numbers satisfying inequality

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October 9, 2010
inequalitiesratioinequalities unsolved

Problem Statement

Prove that there exist distinct natural numbers m1,m2,⋯ ,mkm_1,m_2, \cdots , m_k satisfying the conditions π−1984<25−(1m1+1m2+⋯+1mk)<π−1960\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960} where π\pi is the ratio between a circle and its diameter.
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