We solve problems

Problems Geometry Plane geometry Area Area of a circle, sector, segment

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

Problem #PRU-7653

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

12–14 4.0

In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.

Problem #PRU-78270

Problems Geometry Plane geometry Area Area of curvilinear shapes Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 4.0

120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.

Problem #PRU-88095

Problems Geometry Plane geometry Area

10–12 2.0

A page of a calendar is partially covered by the previous torn sheet (see the figure). The vertices A and B of the upper sheet lie on the sides of the bottom sheet. The fourth vertex of the lower leaf is not visible – it is covered by the top sheet. The upper and lower pages, of course, are identical in size to each other. Which part of the lower page is greater, that which is covered or that which is not?

Problem #PRU-98652

Problems Geometry Plane geometry Area Area ratio Area ratio of similar figures

11–12 1.0

The population of China is one billion people. It would seem that on a map of China with a scale of 1 : 1,000,000 (1 cm : 10 km), it would be possible to fit a million times fewer people than there is in the whole country. However, in fact, not only 1000, but even 100 people will not be able to be placed on this map. Can you explain this contradiction?

Problem #PRU-103947

Problems Geometry Plane geometry Area

10–12 2.0

Find the area of the figures shown below.

Problem #PRU-116301

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Area

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

Problem #PRU-34905

Problems Geometry Plane geometry Area Area of a trapezium Area ratio Area ratio (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

13–14 4.0

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.

Problem #PRU-56747

Problems Geometry Plane geometry Area Area (other)

13–15 2.0

Let \(E\) and \(F\) be the midpoints of the sides \(BC\) and \(AD\) of the parallelogram \(ABCD\). Find the area of the quadrilateral formed by the lines \(AE, ED, BF\) and \(FC\), if it is known that the area \(ABCD\) is equal to \(S\).

Problem #PRU-56748

Problems Geometry Plane geometry Area Area (other)

14–14 2.0

A polygon is drawn around a circle of radius \(r\). Prove that its area is equal to \(pr\), where \(p\) is the semiperimeter of the polygon.