HMMT 二月 2012 · 代数 · 第 5 题

英文解析

5. Find all ordered triples ( a, b, c ) of positive reals that satisfy: ⌊ a ⌋ bc = 3 , a ⌊ b ⌋ c = 4, and ab ⌊ c ⌋ = 5, where
   ⌊ x ⌋ denotes the greatest integer less than or equal to x .
   √ √ √ √ √ √
   30 30 2 30 30 30 30
   Answer: ( , , ) , ( , , ) Write p = abc , q = ⌊ a ⌋⌊ b ⌋⌊ c ⌋ . Note that q is an integer.
   3 4 5 3 2 5
   Multiplying the three equations gives:
   √
   p =60
   Substitution into the first equation,qa ⌊ a ⌋ + 1
   p = 3 < 3 ≤ 6
   ⌊ a ⌋ ⌊ a ⌋
   Looking at the last equation:
   c ⌊ c ⌋
   p = 5 ≥ 5 ≥ 5
   ⌊ c ⌋ ⌊ c ⌋
   Here we’ve used ⌊ x ⌋ ≤ x < ⌊ x ⌋ + 1, and also the apparent fact that ⌊ a ⌋ ≥ 1. Now:
   √
   5 ≤ ≤ 660
   12 5 q
   ≥ q ≥
   5 3
   Since q is an integer, we must have q = 2. Since q is a product of 3 positive integers, we must havethose be 1,1, and 2 in some order, so there are three cases:
   √
   √
   Case 1: ⌊ a ⌋ = 2. By the equations, we’d need a = 30 = 120 / 9 > 3, a contradiction, so 2
   there are no solutions in this case.3
   Case 2: ⌊ b ⌋ = 2. We have the solution
   ( )
   √ √ √
   30 30 30
   , ,
   3 2 5
   Case 3: ⌊ c ⌋ = 2. We have the solution
   ( )
   √ √ √
   30 30 2 30
   , ,
   3 4 5
   Algebra Test