HMMT November, 2019

Solution:

Since each piece has area at least $1$ and the original square has area $10000$, Dylan can end up with at most $10000$ pieces. There is initially $1$ piece, so the number of pieces can increase by at most $9999$. Each cut increases the number of pieces by at least $1$, so Dylan can make at most $9999$ cuts. Notice that this is achievable if Dylan makes $9999$ vertical cuts spaced at increments of $\frac{1}{100}$ units.