Question

The number of distinct pairs of integers (m, n) satisfying $|1+mn| < |m+n| < 5$ is

Answer 1

Given:
$|1+mn| < |m+n| < 5$

To find the distinct pairs (m, n) that satisfy the above conditions, we'll tackle each inequality separately.

1. For $|1+mn| < |m+n|$: This inequality is satisfied if either of the following conditions hold:
   a) $1+mn > 0$ and $1+mn < m+n$
   b) $1+mn < 0$ and $1+mn > -(m+n)$

2. For $|m+n| < 5$: This inequality gives us four possible conditions:
   a)$m+n < 5$
   b)$m+n > -5$

c) $-(m+n) < 5$ or $m+n > -5$ (which is the same as the above condition)

d) $-(m+n) > -5$ or $m+n < 5$ (which is also the same as the first condition)

Considering the range for $|m+n|$ which is (-5, 5), we can make a rough estimate:
For m = 0, n can range from -5 to 4.
For m = 1, n can range from -6 to 3.
Similarly, for m = 2, n can range from -7 to 2.
This pattern continues until the value of m+n reaches 5 or m+n reaches -5.

Now, considering the first inequality $|1+mn| < |m+n|$, we'll check for values within our defined range.

By testing pairs, we can derive the following pairs that satisfy both inequalities:
$(0,1), (0,2), (0,3), (0,4), (1,0), (1,-1), (1,-2), (1,-3), (1,2), (1,3), (-1,0), (-1,1), (-1,2),$
$(-1,-3), (2,1), (2,-1), (2,-2), (2,-4), (-2,1), (-2,-1), (-2,2), (-2,-4), (3,0), (3,-1), (3,-2),$
$(3,-5), (-3,0), (-3,-1), (-3,2), (-3,-5), (4,-1), (4,-2), (4,-3), (-4,-1), (-4,2), (-4,-3)$

There are 36 pairs in total that satisfy both inequalities.

So, the number of distinct pairs (m, n) is 36.