Math Gold Medalist

Lor

2023 USAJMO

Problem 3

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don’t overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.

Small Example

Coloring

Considering Range

Induction

Extremal Combinatorics

Invariant

Check:

n=1 -> Ans=1

n=3 -> Ans=4

Color the table with black and white

n=5 -> Ans=9

n=7 -> Ans=16

Ans=((n+1)/2)^2

Consider range

Place 1 in the cells that their row and column is odd

n(1)=((n+1)/2)^2

Place 2 in the cells that their row and column is even

n(1)=((n-1)/2)^2

Prove that if the hole is determined the table is determined

 

2020 IMO Problem 4

2016 IMO Problem 2

2023 BMO Round 2 Problem 2(British Mathematical Olympiad)

2023 BMO Round 2 Problem 3(British Mathematical Olympiad)

   Solution