MathDB

square split into rectangles

Source: 45th International Tournament of Towns, Senior A-Level P3, Fall 2023

12/11/2023

A square was split into several rectangles so that the centers of rectangles form a convex polygon. a) Is it true for sure that each rectangle adjoins to a side of the square? b) Can the number of rectangles equal 23 ? Alexandr Shapovalov

combinatorial geometry

Domino card

Source: 45th International Tournament of Towns, Junior A-Level P3, Fall 2023

12/16/2023

3. Let us call a bi-squared card

2 \times 1

regular, if two positive integers are written on it and the number in the upper square is less than the number in the lower square. It is allowed at each move to change both numbers in the following manner: either add the same integer (possibly negative) to both numbers, or multiply each number by the same positive integer, or divide each number by the same positive integer. The card must remain regular after any changes made. What minimal number of moves is sufficient to get any regular card from any other regular card? Alexey Glebov

combinatorics

100-digit positive integers

Source: 45th International Tournament of Towns, Senior O-Level P3, Fall 2023

12/16/2023

3. Consider all 100-digit positive integers such that each decimal digit of these equals

2,3,4,5,6

, or 7 . How many of these integers are divisible by

2^{100}

? Pavel Kozhevnikov

number theory

farmers and a crown

Source: 45th International Tournament of Towns, Junior O-Level P3, Fall 2023

12/16/2023

3. Eight farmers have a checkered

8 \times 8

field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 8 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 9 berries from the field so that for sure no farmer will notice this? Tatiana Kazitsyna

combinatorics

3

Problems(4)