74th Romanian Mathematical Olympiad

For simplicity we will call the numbers red, yellow, respectively blue, and a sequence of $9$ consecutive numbers in which the first two are red, the next three are yellow, and the last four are blue will be called a complete coloring.

a) Since $2025=M_9$, the last four numbers are blue, hence $2024$ is blue.

b) The $k+1$-th complete coloring (where $k$ is a natural number), assigns yellow to the numbers $3+9\cdot k$, $4+9\cdot k$ and $5+9\cdot k$, with sum $12+27\cdot k$. After $12$, respectively $13$ complete colorings, the sum of the yellow numbers is $12\cdot 12+27\cdot (0+1+2+\cdots +11)=1926$, respectively $12\cdot 13+27\cdot (0+1+2+\cdots +12)=2262$. In order to get the sum of the yellow numbers larger than $2024$, we need $12$ complete colorings ($108$ numbers), then the red numbers $109$, $110$ and the yellow number $111$, hence the smallest $n$ is $111$.