1. Vasya claims that he drew a rectangle on graph paper that can be cut along the cell edges into one strip of $1 \times 37$ cells and 135 three-cell corners. Is Vasya correct?
Answer: Vasya is wrong.

Solution. If such a rectangle existed, its area would be $37+135 \cdot 3=$ 442 cells. To be able to cut out a strip $1 \times 37$ from the rectangle, one of its sides must be at least 37. Since $442=2 \times 13 \times 17$, the only suitable rectangles are $1 \times 442$ and $2 \times 221$. However, it is impossible to cut out any corner from the first one, and after cutting out the strip $1 \times 37$ from the second one, a section remains from which it is impossible to cut out any corner.

Comment. Answer without justification - 0 points. Area of the drawn rectangle found - 1 point. Some specific examples of rectangles considered - no more than 3 points.