MathDB

Let

A,B

be two points in the plane with integer coordinates

A=(x_1,y_1)

and

B=(x_2,y_2)

. (Thus

x_i,y_i\in\mathbb Z

, for

i=1,2

.) A path

\pi:A\to B

is a sequence of down and right steps, where each step has an integer length, and the initial step starts from

A

, the last step ending at

B

. In the figure below, we indicated a path from

A_1=(4,9)

B1=(10,3)

. The distance

d(A,B)

between

A

and

B

is the number of such paths. For example, the distance between

A=(0,2)

and

B=(2,0)

equals

6

. Consider now two pairs of points in the plane

A_i=(x_i,y_i)

and

B_i=(u_i,z_i)

for

i=1,2

, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):

x_2<x_1

and

y_1>y_2

, which means that

A_1

is North-East of

A_2

;

u_2<u_1

and

z_1>z_2

, which means that

B_1

is North-East of

B_2

. Each of the points

A_i

is North-West of the points

B_j

, for

1\le i

j\le2

. In terms of inequalities, this means that

x_i<\min\{u_1,u_2\}

and

y_i>\max\{z_1,z_2\}

for

i=1,2

.
https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi9hL2I4ODlmNDAyYmU5OWUyMzVmZmEzMTY1MGY3YjI3YjFlMmMxMTI2LnBuZw==&rn=VlRSTUMgMjAxNC5wbmc=
(a) Find the distance between two points

A

and

B

as before, as a function of the coordinates of

A

and

B

. Assume that

A

is North-West of

B

. (b) Consider the

2\times2

matrix

M=\begin{pmatrix}d(A_1,B_1)&d(A_1,B_2)\\d(A_2,B_1)&d(A_2,B_2)\end{pmatrix}

. Prove that for any configuration of points

A_1,A_2,B_1,B_2

as described before,

\det M>0

Problem Statement