Problems

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 \(\frac{1}{9}\).

The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.

At a factory known to us, we cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and therefore the standard deviation of the radius is 10 mm. Engineer Gavin believes that a stack of 100 disks on average will weigh 10,000 kg. By how much is the engineer Gavin wrong?

On weekdays, the Scattered Scientist goes to work along the circle line on the London Underground from Cannon Street station to Edgware Road station, and in the evening he goes back (see the diagram).

Entering the station, the Scientist sits down on the first train that arrives. It is known that in both directions the trains run at approximately equal intervals, and along the northern route (via Farringdon) the train goes from Cannon Street to Edgware Road or back in 17 minutes, and along the southern route (via St James Park) – 11 minutes. According to an old habit, the scientist always calculates everything. Once he calculated that, from many years of observation:

– the train going counter-clockwise, comes to Edgware Road on average 1 minute 15 seconds after the train going clockwise arrives. The same is true for Cannon Street.

– on a trip from home to work the Scientist spends an average of 1 minute less time than a trip home from work.

Find the mathematical expectation of the interval between trains going in one direction.

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.

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?

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.

The area of the red triangle is \(25\) and the area of the orange triangle is \(49\). What is the area of the trapezium \(ABCD\)?

Prove that the relation between areas of two similar polygons equals to the square of their similarity coefficient.