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Problems Geometry Plane geometry Area Area of a polygon Finding the area by rearrangment

The figure below is a regular pentagram. What is larger, the black area or the blue area?

Problems Geometry Plane geometry Area Finding the area by rearrangment

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$ .

Problem #PRU-100111

Problems Geometry Plane geometry Area Area (other) Finding the area by rearrangment

11–13 4.0

In a parallelogram $A B C D$ , point $E$ belongs to the side $A B$ , point $F$ belongs to the side $C D$ and point $G$ belongs to the side $A D$ . What is more, the marked red segments $A E$ and $C F$ have equal lengths. Prove that the total grey area is equal to the total black area.

Problem #PRU-58107

Problems Geometry Plane geometry Area Finding the area by rearrangment Discrete Mathematics 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.