## Mathematics Days in Sofia

### An extraordinary edition of the competition "Mathematics with Computer"

##### In the rectangles for the answers (always decimal numbers), both a decimal point and a decimal comma can be used.

Problem 1
Two-thirds of the volume of a closed cylindrical can of radius 5 cm is filled with some liquid.
What is the height of the liquid, if the can is laid horizontally?
Record the answer to the nearest hundredth of a centimeter.

Auxiliary file in GeoGebra format.

Problem 2
What part of the square (in percent) is colored green?
Record the answer to the nearest tenth of a percent.

Auxiliary file in GeoGebra format.

Problem 3
What is the largest possible area of a rectangle with sides parallel to the coordinate axes, a vertex at the origin of the coordinate system О(0 , 0) and an opposite vertex lying on a circle with radius 1 and center at point A(4 , 3)?
Record the answer to the nearest hundredth.

Auxiliary file in GeoGebra format.

Problem 4
For a triangle ABC with an area of 13 cm2, it is known that the angles at the vertices A and B are 60 and 70 degrees correspondingly. Find the radius of the circle circumscribing the triangle and record the answer to the nearest hundredth.

Problem 5
A company producing plastic containers got an order for a large number of identical cylindrical containers (with a bottom, but without a lid) each with capacity (volume) of 1 liter. In order to save material, the manufacturer wants to make the surface of the container as small as possible. What should be the height of the container (in centimeters)?
Record the answer to the nearest hundredth of a centimeter

Auxiliary file in GeoGebra format.