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Problems Geometry Plane geometry Area Finding the area by rearrangment Set theory and logic Set theory Inclusion-exclusion principle Methods Pigeonhole principle Pigeonhole principle (area and volume) Proof by contradiction

14–15 3.0

a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.

b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than $\frac{1}{9}$ .

Problem #PRU-7653

Problems Geometry Plane geometry Area Area ratio Finding the area by rearrangment

12–14 4.0

In a pentagon $A B C D E$ , diagonal $A D$ is parallel to the side $B C$ and the diagonal $C E$ is parallel to the side $A B$ . Show that the areas of the triangles $△ A B E$ and $△ B C D$ are the same.