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 $ak=ab{k}^{\prime }=2022k^{\prime }$ for some integer $k^{\prime }$. 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.