HMMT 十一月 2016 · THM 赛 · 第 7 题

HMMT November 2016 — THM Round — Problem 7

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.

解析

Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon. Proposed by: Allen Liu Answer: 42 Consider the vectors corresponding to the sides of the heptagon, and call them [ x , y ] for i between i i ∑ ∑ ∑ 2 2 2 1 and 7. Then since x = y = 0, and a has the same parity as a , we have that x + y i i i i √ 2 2 must be an even number. A side length of a lattice valued polygon must be expressible as a + b , √ √ √ √ √ √ √ so the smallest possible values are 1 , 2 , 4 , 5 , 8 , 9 , 10. However, using the seven smallest √ lengths violates the parity constraint. If we try 13, we indeed can get a heptagon with lengths √ √ √ √ √ √ √ 1 , 2 , 4 , 5 , 8 , 9 , 13. One example is the heptagon (0 , 0) , (3 , 0) , (5 , 1) , (6 , 2) , (3 , 4) , (2 , 4) , (0 , 2), and its sum of squares of side lengths is 1 + 2 + 4 + 5 + 8 + 9 + 13 = 42 .