Math Gold Medalist

Lor

2023 USAMO 

Problem 5

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

Small Example

Modular Arithmetic

Arithmetic Sequence

Consider:

n=3

n=4

n=6

Find a counter example for n=2k

Find a counter example for n=3k

Find a counter example for composite numbers.

check n=5

Consider n=prime and numbers on the table mod p.

2020 IMO Problem 4

2023 BMO Round 2 Problem 2(British Mathematical Olympiad)

2023 BMO Round 2 Problem 3(British Mathematical Olympiad)

   Solution