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Found problems: 2

2019 India PRMO, 22
Tags: telescoping sums
What is the greatest integer not exceeding the sum $\sum^{1599}_{n=1} \dfrac{1}{\sqrt{n}}$?

1992 USAMO, 2
Tags: telescoping sums , trigonometry , induction , algebra
Prove \[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. \]