MathDB

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are

A,B,C

and

D

. Similarly, the small rectangles on the right hand side have areas

A^\prime,B^\prime,C^\prime

and

D^\prime

. It is known that

A\leq A^\prime

B\leq B^\prime

C\leq C^\prime

but

D\leq B^\prime

. [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.68,ymax=6.3; draw((0,3)--(0,0)); draw((3,0)--(0,0)); draw((3,0)--(3,3)); draw((0,3)--(3,3)); draw((2,0)--(2,3)); draw((0,2)--(3,2)); label("

A

",(0.86,2.72),SE*lsf); label("

B

",(2.38,2.7),SE*lsf); label("

C

",(2.3,1.1),SE*lsf); label("

D

",(0.82,1.14),SE*lsf); draw((5,2)--(11,2)); draw((5,2)--(5,0)); draw((11,0)--(5,0)); draw((11,2)--(11,0)); draw((8,0)--(8,2)); draw((5,1)--(11,1)); label("

A'

",(6.28,1.8),SE*lsf); label("

B'

",(9.44,1.82),SE*lsf); label("

C'

",(9.4,0.8),SE*lsf); label("

D'

",(6.3,0.86),SE*lsf); dot((0,3),linewidth(1pt)+ds); dot((0,0),linewidth(1pt)+ds); dot((3,0),linewidth(1pt)+ds); dot((3,3),linewidth(1pt)+ds); dot((2,0),linewidth(1pt)+ds); dot((2,3),linewidth(1pt)+ds); dot((0,2),linewidth(1pt)+ds); dot((3,2),linewidth(1pt)+ds); dot((5,0),linewidth(1pt)+ds); dot((5,2),linewidth(1pt)+ds); dot((11,0),linewidth(1pt)+ds); dot((11,2),linewidth(1pt)+ds); dot((8,0),linewidth(1pt)+ds); dot((8,2),linewidth(1pt)+ds); dot((5,1),linewidth(1pt)+ds); dot((11,1),linewidth(1pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e.

A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime

Problem Statement