HMMT 十一月 2016 · THM 赛 · 第 7 题

英文解析

7. Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimumpossible 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 betweeni i
   ∑ ∑ ∑
   2 2 2
   1 and 7. Then since x = y = 0, and a has the same parity as a , we have that x + yi ii 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 .