HMMT 二月 2016 · 冲刺赛 · 第 32 题

HMMT February 2016 — Guts Round — Problem 32

[ 16 ] How many equilateral hexagons of side length 13 have one vertex at (0 , 0) and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2016, February 20, 2016 — GUTS ROUND Organization Team Team ID#

解析

[ 16 ] How many equilateral hexagons of side length 13 have one vertex at (0 , 0) and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but 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 cases are 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, con- tributing 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 .