Question

Find the number of integer pairs (x, y) that satisfy the following system of inequalities:
\[ \begin{cases} x \geq y \geq 3 \\ x + y \leq 14 \end{cases} \]

Hint:

Sketch the triangular region bounded by x=y, y=3 and x+y=14 first, seeing the three vertices makes it easier to decide which variable to fix and what range it should take, before you start counting lattice points.

Answer 1

Correct Answer: 25

Step 1: The region is a triangle: \(y \geq 3\), \(x \geq y\), and \(x+y\leq14\), with corners at \((3,3)\), \((11,3)\), and \((7,7)\).
Step 2: Count lattice points row by row along \(y\). Since \(x\) runs from \(y\) to \(14-y\), each row \(y\) has \(15-2y\) points, valid only while \(15-2y\geq1\), i.e. \(y\leq7\).
Step 3: This is an arithmetic sequence \(9,7,5,3,1\) as \(y\) goes \(3\to7\) (5 terms, first \(9\), last \(1\)). Its sum is \(\dfrac{5}{2}(9+1)=25\).
Final Answer: \[ \boxed{25} \]