Subject 4 Consider the table formed by pairs of natural numbers

![](https://cdn.mathpix.com/cropped/2024_06_07_de8b632b716bedecf3ccg-3.jpg?height=303&width=558&top_left_y=1047&top_left_x=773)

a) What is the pair in the middle of the 9th row?

b) What is the 1000th pair on the 2010th row?

c) In how many pairs written from row 1 to row 2010 inclusive, does the number 1005 appear?
## Solution and Grading

a) Write the 9th row:

$(1,9) ;(2,8) ;(3,7) ;(4,6) ;(5,5) ;(6,4) ;(7,3) ;(8,2) ;(9,1)$

$1 \text{p}$

The pair is $(5,5)$

b) Write row 2010

$(1,2010),(2,2009),(3,2008),(4,2007),(5,2006), \ldots,(1000,1011), \ldots .(2010,1) \quad 1 \text{p}$

The required pair is $(1000,1011)$

$1 \text{p}$

c) Row 1005

$(1,1005),(2,1004),(3,1003), \ldots,(1005,1),(1004,2) \ldots \Rightarrow 2$ pairs

Row 1006

$$
(1,1006),(2,1005),(3,1004), \ldots,(1005,2),(1006,1) \ldots \Rightarrow 2 \text{ pairs }
$$

Row 1007

$$
(1,1007),(2,1006),(3,1005), \ldots,(1005,3),(1006,2),(1007,1) \Rightarrow 2 \text{ pairs }
$$

$\qquad$

![](https://cdn.mathpix.com/cropped/2024_06_07_de8b632b716bedecf3ccg-3.jpg?height=512&width=277&top_left_y=2180&top_left_x=1577)

Row 2009

$(1,2009),(2,2008),(3,2007), \ldots,(1005,1005),(1006,1004), \ldots,(2009,1) \Rightarrow 1$ pair $1 \text{p}$

Row 2010

$(1,2010),(2,2009),(3,2008), \ldots(1005,1006),(1006,1005),(1007,1004), \ldots,(2010,1)$

$=2$ pairs

The number 1005 appears in:

$(2010-1004): 2-1=1006 \cdot 2-1=2012-1=2011$ pairs

$1 \text{p}$

Note: Any other correct solution or correct attempt at solving will be graded accordingly.