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.

Answer 2

\(|1 + mn| < |m + n| < 5\)
For two numbers ‘a’ and ‘b’,
\(|a| < |b| \)is equivalent to \(a^2 < b^2\)


So, we can say that:
\((1 + mn)^2 < (m + n)^2\)
\(1 + 2mn +m^2n^2 < m^2 + n^2 + 2mn\)
\(1 - n^2 - m^2 + m^2n^2 < 0\)
\((1 - n^2) - m^2(1 - n^2) < 0\)
\((1 - m^2)(1 - n^2) < 0\)


For the product to be negative, either one of the two terms has to be negative.
But they cannot simultaneously be 0.
The only possibility for either of the two terms to be positive is when
\(n = 0 \) and \(|m| > 1\), or \(|n| > 1\) and \(m = 0\)


Now for the case when \(m = 0 \) and \(|n| > 1\)
\(|m + n| < 5\)
\(|0 + n| < 5\)
So n can be \(±2, ±3, ±4\)
Which are 6 cases

Similarly for the case when \(n = 1\) and \(|m| > 1\)
\(|m + n| < 5\)
\(|0 + m| < 5\)
So m can be 
Again we have 6 cases.
Hence the answer is 12.