The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set:

For a set of 9 observations, the median is the $\left(\frac{9+1}{2}\right)=5\text{th}$ observation when arranged in ascending order.
Let the sorted observations be $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9$.
Median $x_5=20.5$.
The largest 4 observations are $x_6,x_7,x_8,x_9$.
If these are increased, they still remain larger than $x_5$.
The value and position of the 5th observation ($x_5$) do not change.
Therefore, the median remains the same.