HMMT 十一月 2020 · GEN 赛 · 第 1 题
HMMT November 2020 — GEN Round — Problem 1
题目详情
- In the Cartesian plane, a line segment with midpoint (2020 , 11) has one endpoint at ( a, 0) and the other endpoint on the line y = x . Compute a . √
解析
- In the Cartesian plane, a line segment with midpoint (2020 , 11) has one endpoint at ( a, 0) and the other endpoint on the line y = x . Compute a . Proposed by: Lingyi Qiu Answer: 4018 ( ) a + t t Solution: Let the other endpoint be ( t, t ). The midpoint of ( a, 0) and ( t, t ) is , . So, we know 2 2 a + t t that = 2020 and = 11. The second equation yields t = 22. Substituting this into the first yields 2 2 a = 2 · 2020 − 22 = 4018. √