11.2. On a coordinate plane, a grasshopper jumps starting from the origin. The first jump, one cm long, is directed along the OX axis, each subsequent jump is 1 cm longer than the previous one, and is directed perpendicular to the previous one in either of two directions at its choice. Can the grasshopper end up at the origin after the hundredth jump?
Answer: No.

Solution: The grasshopper will make 50 vertical and 50 horizontal jumps. The lengths of the vertical jumps are 2, 4, ..., 100 cm. Among these numbers, 25 are divisible by 4 and 25 give a remainder of 2 when divided by 4. Therefore, regardless of the arrangement of plus and minus signs between them, the sum will always give a remainder of 2 when divided by 4 and will never be equal to 0. The grasshopper will not be able to return to the x-axis after 100 jumps, and thus, will not be able to return to the origin.

Criteria: The idea of considering horizontal and vertical jumps separately is present: 1 point. Lack of proof that the sum of the lengths of the vertical jumps will never be equal to 0 regardless of the arrangement of plus and minus signs: minus 3 points.