We solve problems

Problems Geometry Plane geometry Area Triangle area

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:

Problem #PRU-100108

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

11–13 4.0

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

Problem #PRU-100110

Problems Geometry Plane geometry Area Finding the area by rearrangment

11–13 4.0

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 \(ABCD\), point \(E\) belongs to the side \(AB\), point \(F\) belongs to the side \(CD\) and point \(G\) belongs to the side \(AD\). What is more, the marked red segments \(AE\) and \(CF\) have equal lengths. Prove that the total grey area is equal to the total black area.

Problem #PRU-100112

Problems Algebra and arithmetic Arithmetics. Mental maths Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

10–10

Replace the letters with digits in a way that makes the following sum as big as possible: \[SEND +MORE +MONEY.\]

Problem #PRU-100113

Problems Algebra and arithmetic Arithmetics. Mental maths Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

Problem #PRU-100114

Problems Algebra and arithmetic Arithmetics. Mental maths Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Problem #PRU-100202

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(4\times 6\) rectangle?

Problem #PRU-100203

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(5\times8\) rectangle.

Problem #PRU-100204

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

Having mastered tiling small rooms, Robinson wondered if he could tile big spaces, and possibly very big spaces. He wondered if he could tile the whole plane. He started to study the tiling, which can be continued infinitely in any direction. Can you help him with it?

Tile the whole plane with the following shapes: