2023/2024 Tournament of Towns

Part of Tournament Of Towns

Subcontests

(7)

2

three hands on a clock.

2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

square or not necessarily a square

2. The quadrilateral

A B C D

is convex. Its sides

A B

C D

are parallel. It is known that the angles

D A C

A B D

are equal. Furthermore the angles

C A B

D B C

A B C D

necessarily a square? Alexandr Terteryan

2

A $2 \times 2024$ table

2 \times 2024

is filled with positive integers. Specifically, the first row is filled with numbers from the set

\{1, \ldots, 2023\}

. It turned out that for any two columns the difference of numbers from the first row is divisible by the difference of numbers from the second row, while all numbers in the second row are pairwise different. Is it true for sure that the numbers in the first row are equal? Ivan Kukharchuk

rectangles cutting

6. The baker has baked a rectangular pancake. He then cut it into

n^{2}

rectangles by making

n-1

n-1

vertical cuts. Being rounded to the closest integer, the areas of resulting rectangles equal to all positive integers from 1 to

n^{2}

in some order. For which maximal

n

could this happen? (Half-integers are rounded upwards.) Georgy Karavaev

4

4

4

4

1

Weighings using a balance

On the table there are

2n

coins that look the same. It is known that

n

of them weigh 9 g. each, while the remaining

n

weigh 10 g. each. It is required to split the coins into

n

pairs with total weight of each pair 19 g. Prove that this can be done in less than

n

weighings using a balance without additional weights (the balance shows which pan is heavier or that their weight is equal).