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:

When counting integer points defined by linear inequalities, fix one variable and find the range for the second. Summing the number of possibilities for each fixed value gives the total count.

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: \[ \boxed{25} \]