IMO Shortlist 1996 problem C1

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Dodao/la: arhiva
2. travnja 2012.
We are given a positive integer r and a rectangular board ABCD with dimensions AB = 20, BC = 12. The rectangle is divided into a grid of 20 \times 12 unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is \sqrt {r}. The task is to find a sequence of moves leading from the square with A as a vertex to the square with B as a vertex.

(a) Show that the task cannot be done if r is divisible by 2 or 3.

(b) Prove that the task is possible when r = 73.

(c) Can the task be done when r = 97?
Izvor: Međunarodna matematička olimpijada, shortlist 1996