HMMT 二月 2024 · 冲刺赛 · 第 22 题
HMMT February 2024 — Guts Round — Problem 22
题目详情
- [12] Let x < y be positive real numbers such that √ √ √ √ x + y = 4 and x + 2 + y + 2 = 5 . Compute x .
解析
- [12] Let x < y be positive real numbers such that √ √ √ √ x + y = 4 and x + 2 + y + 2 = 5 . Compute x . Proposed by: Ethan Liu 49 Answer: 36 Solution: Adding and subtracting both equations gives √ √ √ √ x + 2 + x + y + 2 + y = 9 √ √ √ √ x + 2 − x + y + 2 − y = 1 √ √ √ √ √ √ √ √ Substitute a = x + x + 2 and b = y + y + 2 . Then since ( x + 2 + x )( x + 2 − x ) = 2 , we have a + b = 9 2 2
- = 1 a b Dividing the first equation by the second one gives ab = 18 , a = 3 , b = 6 √ √ √ √ 2 √ 3 − x +2+ x − ( x +2 − x ) 7 49 3 Lastly, x = = = , so x = . 2 2 6 36