HMMT 十一月 2018 · GEN 赛 · 第 3 题
HMMT November 2018 — GEN Round — Problem 3
题目详情
- A square in the xy -plane has area A , and three of its vertices have x -coordinates 2 , 0 , and 18 in some order. Find the sum of all possible values of A .
解析
- A square in the xy -plane has area A , and three of its vertices have x -coordinates 2 , 0 , and 18 in some order. Find the sum of all possible values of A . Proposed by: Yuan Yao Answer: 1168 More generally, suppose three vertices of the square lie on lines y = y , y = y , y = y . One of these 1 2 3 vertices must be adjacent to two others. If that vertex is on y = y and the other two are on y = y 1 2 and y = y , then we can use the Pythagorean theorem to get that the square of the side length is 3 2 2 ( y − y ) + ( y − y ) . 2 1 3 1 2 2 2 2 2 2 2 2 For ( y , y , y ) = (2 , 0 , 18), the possibilities are 2 + 16 , 2 + 18 , 16 + 18 , so the sum is 2(2 + 16 + 1 2 3 2