HMMT 二月 2012 · 代数 · 第 5 题

HMMT February 2012 — Algebra — Problem 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 .

解析

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: √ 60 p = q Substitution into the first equation, a ⌊ 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: √ 60 5 ≤ ≤ 6 q 12 5 ≥ q ≥ 5 3 Since q is an integer, we must have q = 2. Since q is a product of 3 positive integers, we must have those be 1,1, and 2 in some order, so there are three cases: √ √ 2 Case 1: ⌊ a ⌋ = 2. By the equations, we’d need a = 30 = 120 / 9 > 3, a contradiction, so 3 there are no solutions in this case. 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