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number of sets with 2n + 1 points in coordinate plane

Source: KJMO 2011 p4

May 4, 2019
analytic geometrycombinatoricscombinatorial geometrySets

Problem Statement

For a positive integer nn, (n≥2n\ge 2), find the number of sets with 2n+12n + 1 points P0,P1,...,P2nP_0, P_1,..., P_{2n} in the coordinate plane satisfying the following as its elements: - P0=(0,0),P2n=(n,n)P_0 = (0, 0),P_{2n}= (n, n) - For all i=1,2,...,2n−1i = 1,2,..., 2n - 1, line PiPi+1P_iP_{i+1} is parallel to xx-axis or yy-axis and its length is 11. - Out of 2n2n linesP0P1,P1P2,...,P2n−1P2nP_0P_1, P_1P_2,..., P_{2n-1}P_{2n}, there are exactly 44 lines that are enclosed in the domain y≤xy \le x.
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