Estonian Math Competitions

Answer: $2022$.

Let the rectangle be of size $a \times b$. Let there be $k$ black unit squares in every row and $l$ black unit squares in every column; then $ak = bl$. Since $ab = 2022 = 2 \cdot 3 \cdot 337$ where the factors are primes, numbers $a$ and $b$ must be coprime. Thus $b \mid k$, implying $ak = abk' = 2022k'$ for some integer $k'$. Since at least one black square exists and the total number of unit squares is $2022$, the only possibility is $ak = 2022$, i.e., all unit squares are black.