K1) We consider a white $5 \times 5$ square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle?
Solution 1 (Tanish), Count opposite corners: We consider the corners of any rectangle. These lie in a grid consisting of 36 points in a $6 \times 6$ square. We choose one of these points as the first vertex. If we now choose the opposite corner point, we have defined a rectangle. Opposite corners cannot be in the same row or column, so we choose one of 25 other points. In total we have 36 times 25 rectangles. But we have counted each rectangle 4 times - there are two pairs of opposite corners, each of which we have counted twice - so we divide our result by 4 and get $9 \times 25=225$.

Note (Jana): It is also possible to count the possible pairs of opposite vertices in the following way: First we choose two different points from the $6 \times 6$ square. There are $\binom{36}{2}=18 \cdot 35$ possibilities. Now we have to subtract the pairs that are in the same row or column. Since there are a total of 6 rows and 6 columns and we have $\binom{6}{2}$ possibilities to select a pair in each of these columns or rows, we need to subtract $12 \cdot\binom{6}{2}=12 \cdot 15$. In this case, we have counted each rectangle twice, so we get the result $\frac{1}{2} \cdot(18 \cdot 35-12 \cdot 15)=225$.