HMMT 十一月 2010 · 团队赛 · 第 7 题

HMMT November 2010 — Team Round — Problem 7

[ 7 ] ABC is a right triangle with ∠ A = 30 and circumcircle O . Circles ω , ω , and ω lie outside 1 2 3 ABC and are tangent to O at T , T , and T respectively and to AB , BC , and CA at S , S , and S , 1 2 3 1 2 3 ′ ′ ′ respectively. Lines T S , T S , and T S intersect O again at A , B , and C , respectively. What is 1 1 2 2 3 3 ′ ′ ′ the ratio of the area of A B C to the area of ABC ? Linear? What’s The Problem? A function f ( x , x , . . . , x ) is said to be linear in each of its variables if it is a polynomial such that 1 2 n no variable appears with power higher than one in any term. For example, 1 + x + xy is linear in x 2 2 and y , but 1 + x is not. Similarly, 2 x + 3 yz is linear in x , y , and z , but xyz is not. 1

解析

[ 7 ] ABC is a right triangle with ∠ A = 30 and circumcircle O . Circles ω , ω , and ω lie outside 1 2 3 ABC and are tangent to O at T , T , and T respectively and to AB , BC , and CA at S , S , and S , 1 2 3 1 2 3 ′ ′ ′ respectively. Lines T S , T S , and T S intersect O again at A , B , and C , respectively. What is 1 1 2 2 3 3 ′ ′ ′ the ratio of the area of A B C to the area of ABC ? √ 3+1 Answer: Let [ P QR ] denote the area of 4 P QR . The key to this problem is following fact: 2 1 [ P QR ] = P Q · P R sin ∠ QP R . 2 √ √ 3 ◦ Assume that the radius of O is 1. Since ∠ A = 30 , we have BC = 1 and AB = 3. So [ ABC ] = . 2 ′ ′ ◦ ′ ◦ ′ Let K denote the center of O . Notice that ∠ B KA = 90 , ∠ AKC = 90 , and ∠ B KA = ∠ KAB = ◦ ′ ′ ′ ′ ◦ ′ ′ ◦