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HMMT 二月 2013 · 团队赛 · 第 1 题

HMMT February 2013 — Team Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 10 ] Let a and b be real numbers such that = . Find all possible values of . 2 2 2 2 a + b 4 a + b
解析
  1. [ 10 ] Let a and b be real numbers such that = . Find all possible values of . 2 2 2 2 a + b 4 a + b √ 3 Answer: The hypothesis statement is equivalent to 2 2 2 a + b = 4 ab 2 1 : ( a + b ) = 6 ab 2 2 : ( a − b ) = 2 ab Multiplying equations 1 and 2, 2 2 2 2 ( a − b ) = 12( ab ) √ 2 2 | a − b | = ± 12 ab Since the left hand side and ab are both positive, √ 2 2 | a − b | = 12 ab √ √ √ 2 2 | a − b | 12 ab 12 3 = = = 2 2 2 2 a + b a + b 4 2 (It is clear that such a and b exist: for example, we can take a = 1 and solve for b by way of the quadratic formula.)