Question 77
Let $A$ and $B$ be two identical convex polygons, each having area $2015$. The polygon $A$ is divided into polygons $A_1, A_2, \dots, A_{2015}$ with positive area, and the polygon $B$ into polygons $B_1, B_2, \dots, B_{2015}$ with positive area. The polygons $A_1, A_2, \dots, A_{2015}, B_1, B_2, \dots, B_{2015}$ are colored with $2015$ colors, in such a way that $A_i$ is colored differently from $A_j$ and $B_i$ is colored differently from $B_j$, for $i \neq j$. After overlapping the polygons $A$ and $B$, we calculate the sum of the areas of the parts that have the same color.
Prove that there exists a coloring of the polygons for which this sum is at least $1$.
Prove that there exists a coloring of the polygons for which this sum is at least $1$.