Question

The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:

Hint:

When given two-variable inequalities, fix one variable and count valid values of the other. Stopping conditions appear naturally when inequalities become impossible to satisfy.

Answer 1

Correct Answer: 16

Step 1: Shift variables to make them non-negative: let \(y = 3+j\) with \(j \ge 0\), and since \(x > y\), let \(x = y+1+i = 4+j+i\) with \(i \ge 0\).
Step 2: Substitute into \(x+y<14\): \((4+j+i)+(3+j) < 14 \Rightarrow 2j+i < 7 \Rightarrow i \le 6-2j\), which needs \(j \le 3\). For each \(j\), the number of valid \(i\) (from 0 to \(6-2j\)) is \(7-2j\).
Step 3: Sum over \(j=0,1,2,3\): \(7+5+3+1 = 16\).
\[ \boxed{16} \]