6

Part of 2023/2024 Tournament of Towns

Problems(2)

A $2 \times 2024$ table

Source: 45th International Tournament of Towns, Senior A-Level P6, Fall 2023

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

Source: 45th International Tournament of Towns, Junior A-Level P6, Fall 2023

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

rectanglecombinatorial geometry