HMMT 二月 2009 · 代数 · 第 6 题
HMMT February 2009 — Algebra — Problem 6
题目详情
英文原题
- [ 5 ] Let x and y be positive real numbers and θ an angle such that θ 6 = n for any integer n . Supposesin θ cos θ2
x y=
4 andcos θ sin θ 97 sin 2 θ4 + = .
4 4 3 3
x y x y + y xy
Compute + .xy x
解析
英文解析
- [ 5 ] Let x and y be positive real numbers and θ an angle such that θ 6 = n for any integer n . Supposesin θ cos θ2
x y=
4 andcos θ sin θ 97 sin 2 θ4 + = .
4 4 3 3
x y x y + y xy 1
Compute + .xy x
Answer: 4
Solution: From the first relation, there exists a real number k such that x = k sin θ and y = k cos θ .
Then we havecos θ sin θ 194 sin θ cos θ44 + = = 194 .
4 2
cos θ24
sin θ sin θ cos θ (cos θ + sin θ )
4 4
x 2 2 cos θ sin θy
Notice that if t = + then ( t − 2) − 2 = + = 194 and so t = 4.
y x sin θ cos θ44