MEMO 2017 ekipno problem 4

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Dodao/la: arhiva
Sept. 12, 2018

Let n \geqslant 3 be an integer. A sequence P_1, P_2, \ldots, P_n of distinct points in the plane is called good if no three of them are collinear, the polyline P_1P_2 \ldots P_n is non-self-intersecting and the triangle P_iP_{i + 1}P_{i + 2} is oriented counterclockwise for every i = 1, 2, \ldots, n - 2. For every integer n \geqslant 3 determine the greatest possible integer k with the following property: there exist n distinct points A_1, A_2, \ldots, A_n in the plane for which there are k distinct permutations \sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\} such that A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)} is good.
(A polyline P_1P_2 \ldots P_n consists of the segments P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n.)

Source: Srednjoeuropska matematička olimpijada 2017, ekipno natjecanje, problem 4