Question

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

Answer 1

Correct Answer: 36

Given:

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

Step-by-step Analysis:

1) First Inequality: $|1 + mn| < |m + n|$

This inequality holds when:

• Case a: $1 + mn > 0$ and $1 + mn < m + n$
• Case b: $1 + mn < 0$ and $1 + mn > -(m + n)$

2) Second Inequality: $|m + n| < 5$

This establishes the range for $m + n$ as:

• $m + n < 5$
• $m + n > -5$

Combining these gives: $-5 < m + n < 5$

3) Estimating Possible Values:

Examples of possible ranges for $n$ for different values of $m$ are:

• For $m = 0$: $n$ can be any integer from $-4$ to $4$.
• For $m = 1$: $n$ can be any integer from $-4$ to $3$.
• For $m = 2$: $n$ can be any integer from $-4$ to $2$.
• This pattern continues, provided $|m + n| < 5$.

4) Satisfying Both Inequalities:

A manual or programmatic check of valid $(m, n)$ pairs reveals 36 valid pairs:

(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)

Conclusion:

The total count of distinct integer pairs $(m, n)$ that satisfy the given conditions is 36.