Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Prove it United States

Problem:
A grasshopper lives on a coordinate line. It starts off at $1$ . It can jump either $1$ unit or $5$ units either to the right or to the left. However, the coordinate line has holes at all points with coordinates divisible by $4$ (e.g. there are holes at $-4, 0, 4, 8$ etc.), so the grasshopper can not jump to any of those points. Can it reach point $3$ after $2003$ jumps?

Solution

Solution:
Each jump changes the parity of grasshopper's coordinate. After $2003$ jumps the grasshopper will be at an even point on the coordinate line, and therefore can not be at $3$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.