20. How many pairs of natural numbers ( $m, n$ ) exist such that $m, n \leqslant 1000$ and

$$
\frac{m}{n+1}<\sqrt{2}<\frac{m+1}{n} ?
$$

## 9th grade
90.20. Answer. 1706. Hint. The pair ( $m, n$ ) satisfies the inequalities from the condition if and only if the line $y=\sqrt{2 x}$ intersects a cell on the coordinate grid plane, the lower left corner of which has coordinates $(m, n)$.