Combinatorics · Invariants · Colouring · Parity · Grid walks
A mosquito sits on the lower left cell of a rectangular board of format \(2003 \times 2004\). It travels above the board in the following manner: taking off from the cell it occupies, it flies over \(99\) cells and lands on the hundredth one to rest. The line along which it flies need not be straight - it may be broken and may even cross itself - but every step of it runs parallel to an edge of the board. From the cell on which it has landed the mosquito takes off again, flies over \(99\) cells, lands on the hundredth, and so on.
Can the mosquito ever land on the upper right cell of the board?
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Serbian Municipal Competition 2004, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source