We solve problems

Problem #PRU-30649

Problems Algebra Algebraic equations and systems of equations Diophantine equations

11–13 1.0

Determine all integer solutions of the equation \(3x - 12y = 7\).

Problem #PRU-35282

Problems Algebra Algebraic equations and systems of equations

12–14

Solve the equation \(xy = x + y\) in integers.

Problem #PRU-89953

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other) Methods Pigeonhole principle Pigeonhole principle (other)

10–12 1.0

In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

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

Tile the whole plane with the following shapes:

Problem #PRU-100205

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

David Smith cut out 12 nets. He claimed that it was possible to make a cube out of each net. Roger Penrose looked at the patterns, and after some considerable thought decided that he was able to make cubes from all the nets except one. Can you figure out which net cannot make a cube?