HMMT 十一月 2011 · 团队赛 · 第 10 题
HMMT November 2011 — Team Round — Problem 10
题目详情
- [ 8 ] Let G G G be a triangle with G G = 7, G G = 13, and G G = 15. Let G be a point 1 2 3 1 2 2 3 3 1 4 − − − → outside triangle G G G so that ray G G cuts through the interior of the triangle, G G = G G , 1 2 3 1 4 3 4 4 2 ◦ and ∠ G G G = 30 . Let G G and G G meet at G . Determine the length of segment G G . 3 1 4 3 4 1 2 5 2 5
解析
- [ 8 ] Let G G G be a triangle with G G = 7, G G = 13, and G G = 15. Let G be a point 1 2 3 1 2 2 3 3 1 4 − − − → outside triangle G G G so that ray G G cuts through the interior of the triangle, G G = G G , 1 2 3 1 4 3 4 4 2 ◦ and ∠ G G G = 30 . Let G G and G G meet at G . Determine the length of segment G G . 3 1 4 3 4 1 2 5 2 5 169 Answer: 23 G 5 y x G 4 G 2 13 7 G G 1 3 15 We first show that quadrilateral G G G G is cyclic. Note that by the law of cosines, 1 2 4 3 Team Round 2 2 2 7 + 15 − 13 1 cos ∠ G G G = = , 2 1 3 2 · 7 · 15 2 ◦ ◦ so ∠ G G G = 60 . However, we know that ∠ G G G = 30 , so G G is an angle bisector. Now, let 2 1 3 3 1 4 1 4 ̂ ̂ G G intersect the circumcircle of triangle G G G at X . Then, the minor arcs G X and G X are 1 4 1 2 3 2 3 subtended by the equal angles ∠ G G X and ∠ G G X , implying that G X = G X , i.e. X is on the 2 1 3 1 2 3 perpendicular bisector of G G , l . Similarly, since G G = G G , G lies on l . However, since l and 2 3 4 2 4 3 4 G G are distinct (in particular, G lies on G G but not l ), we in fact have X = G , so G G G G 1 4 1 1 4 4 1 2 4 3 is cyclic. We now have G G G ∼ G G G since G G G G is cyclic. Now, we have ∠ G G G = ∠ G G G = 5 2 4 5 3 1 1 2 4 3 4 3 2 4 1 2 √ ◦ 30 , and we may now compute G G = G G = 13 / 3. Let G G = x and G G = y . Now, from 2 4 4 3 5 2 5 4 G G G ∼ G G G , we have: 5 4 2 5 1 3 √ x 13 / 3 y √ = = . 15 x + 7 y + 13 / 3 Equating the first and second expressions and cross-multiplying, we get √ √ 13 3 15 3 x y + = . 3 13 Now, equating the first and third expressions and and substituting gives ( ) ( ) √ √ √ 15 3 x 13 3 15 3 x − = x ( x + 7). 13 3 13 Upon dividing both sides by x , we obtain a linear equation from which we can solve to get x = 169 / 23. Team Round