HMMT 十一月 2012 · 冲刺赛 · 第 17 题
HMMT November 2012 — Guts Round — Problem 17
题目详情
- [ 10 ] Given that x = ln 30, y = ln 360, and z = ln 270, and that there are rational numbers p, q, r such that ln 5400 = px + qy + rz , find the ordered triple ( p, q, r ).
解析
- [ 10 ] ( ) 5 1 1 Answer: , , We can rewrite the equation in terms of ln 2 , ln 3 , ln 5, to get 4 2 4 3 ln 2 + 3 ln 3 + 2 ln 5 = ln 5400 = px + qy + rz = ( p + 3 q + r ) ln 2 + ( p + 2 q + 3 r ) ln 3 + ( p + q + r ) ln 5 . Consequently, since p, q, r are rational we want to solve the system of equations p + 3 q + r = 3 , p + ( ) 5 1 1 2 q + 3 r = 3 , p + q + r = 2, which results in the ordered triple , , . 4 2 4