ILL 1984 Problem 35

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