HMMT 二月 2017 · 冲刺赛 · 第 30 题

HMMT February 2017 — Guts Round — Problem 30

[ 17 ] Consider an equilateral triangular grid G with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, . . . , and 20 points in the last row, for a total of 210 points. Let S be a closed non-self- intersecting polygon which has 210 vertices, using each point in G exactly once. Find the sum of all possible values of the area of S .

解析

[ 17 ] Consider an equilateral triangular grid G with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, . . . , and 20 points in the last row, for a total of 210 points. Let S be a closed non-self- intersecting polygon which has 210 vertices, using each point in G exactly once. Find the sum of all possible values of the area of S . Proposed by: Sammy Luo √ Answer: 52 3 Imagine deforming the triangle lattice such that now it looks like a lattice of 45-45-90 right triangles 2 √ with legs of length 1. Note that by doing this, the area has multiplied by , so we need to readjust 3 √ 3 out answer on the isosceles triangle lattice by a factor of at the end. By Pick’s Theorem, the area 2 √ √ P 3 in the new lattice is given by I + − 1 = 0 + 105 − 1 = 104 . Therefore, the answer is 104 · = 52 3 . 2 2