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HMMT 二月 2010 · GEN1 赛 · 第 1 题

HMMT February 2010 — GEN1 Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 3 ] Suppose that x and y are positive reals such that 2 2 4 x − y = 3 , x + y = 13 . Find x .
解析
  1. [ 3 ] Suppose that x and y are positive reals such that 2 2 4 x − y = 3 , x + y = 13 . Find x . √ 3+ 17 2 2 4 2 Answer: Squaring both sides of x − y = 3 gives x + y − 2 xy = 9. Subtract this equation 2 2 2 4 2 from twice the second given to get x + 2 xy + y = 17 = ⇒ x + y = ± 17. Combining this equation √ √ 3 ± 17 3+ 17 with the first given, we see that x = . Since x is a positive real, x must be . 2 2