74th Romanian Mathematical Olympiad

There are $155$ multiples of $13$ not larger than $2024$: $13\cdot 1$, $13\cdot 2$, $\dots$, $13\cdot 155$.

There are $20$ multiples of $100$ not larger than $2024$: $100\cdot 1$, $100\cdot 2$, $\dots$, $100\cdot 20$.

There is only one common multiple of $13$ and $100$ not larger than $2024$, namely $1300$.

There are $155+20-1=174$ numbers not larger than $2024$, that are multiples of $13$ or multiples of $100$. This leaves $2024-174=1850$ non-winners.

We are sure that we have obtained a winner as soon as we pick $1850+1=1851$ cards.