Question

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

Answer 1

Correct Answer: 36

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.