MathDB

A

and

B

are two identical convex polygons, each with an area of

2015

. The polygon

A

is divided into polygons

A_1, A_2,...,A_{2015}

, while

B

is divided into polygons

B_1, B_2,...,B_{2015}

. Each of these smaller polygons has a positive area. Furthermore,

A_1, A_2,...,A_{2015}

and

B_1, B_2,...,B_{2015}

are colored in

2015

distinct colors, such that

A_i

and

A_j

are differently colored for every distinct

i

and

j

and

B_i

and

B_j

are also differently colored for every distinct

i

and

j

. After

A

and

B

overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least

1

Problem Statement