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

We need integer pairs \((x, y)\) such that: \[ y \ge 3,\quad x>y,\quad x + y<14. \] We check possible integer values of \(y\): Case 1: \(y = 3\) Condition: \[ x>3,\quad x + 3<14 \Rightarrow x<11. \] So possible \(x = 4,5,6,7,8,9,10\). Count: \(7\) pairs. Case 2: \(y = 4\) Condition: \[ x>4,\quad x + 4<14 \Rightarrow x<10. \] So possible \(x = 5,6,7,8,9\). Count: \(5\) pairs. Case 3: \(y = 5\) Condition: \[ x>5,\quad x + 5<14 \Rightarrow x<9. \] So possible \(x = 6,7,8\). Count: \(3\) pairs. Case 4: \(y = 6\) Condition: \[ x>6,\quad x + 6<14 \Rightarrow x<8. \] So possible \(x = 7\). Count: \(1\) pair. Case 5: \(y = 7\) Condition: \[ x>7,\quad x + 7<14 \Rightarrow x<7. \] No solution. Count: \(0\). Total pairs: \[ 7 + 5 + 3 + 1 = 16. \] Thus, the number of distinct integer pairs \((x, y)\) is: \[ \boxed{16}. \]