HMMT 二月 2016 · 冲刺赛 · 第 32 题

英文解析

32. [ 16 ] How many equilateral hexagons of side length 13 have one vertex at (0 , 0) and the other fivevertices at lattice points?
    (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concavebut not self-intersecting.)
    Proposed by: Casey Fu
    Answer: 216
    We perform casework on the point three vertices away from (0 , 0). By inspection, that point can be
    ( ± 8 , ± 3), ( ± 7 , ± 2), ( ± 4 , ± 3), ( ± 3 , ± 2), ( ± 2 , ± 1) or their reflections across the line y = x . The casesare as follows:
    If the third vertex is at any of ( ± 8 , ± 3) or ( ± 3 , ± 8), then there are 7 possible hexagons. There are 8
    points of this form, contributing 56 hexagons.
    If the third vertex is at any of ( ± 7 , ± 2) or ( ± 2 , ± 7), there are 6 possible hexagons, contributing 48
    hexagons.
    If the third vertex is at any of ( ± 4 , ± 3) or ( ± 3 , ± 4), there are again 6 possible hexagons, contributing
    48 more hexagons.
    If the third vertex is at any of ( ± 3 , ± 2) or ( ± 2 , ± 3), then there are again 6 possible hexagons, contributing 48 more hexagons.
    Finally, if the third vertex is at any of ( ± 2 , ± 1), then there are 2 possible hexagons only, contributing
    16 hexagons.
    Adding up, we get our answer of 216 .