HMMT 十一月 2011 · 冲刺赛 · 第 29 题

HMMT November 2011 — Guts Round — Problem 29

[ 14 ] Let ABC be a triangle with AB = 4, BC = 8, and CA = 5. Let M be the midpoint of BC , and let D be the point on the circumcircle of ABC so that segment AD intersects the interior of ABC , and ∠ BAD = ∠ CAM . Let AD intersect side BC at X . Compute the ratio AX/AD .

解析

[ 14 ] Let ABC be a triangle with AB = 4, BC = 8, and CA = 5. Let M be the midpoint of BC , and let D be the point on the circumcircle of ABC so that segment AD intersects the interior of ABC , and ∠ BAD = ∠ CAM . Let AD intersect side BC at X . Compute the ratio AX/AD . 9 Answer: Let E be the intersection of AM with the circumcircle of ABC . We note that, by 41 equal angles ADC ∼ ABM , so that AB 20 AD = AC ( ) = . AM AM Using the law of cosines on ABC , we get that 2 2 2 4 + 8 − 5 55 cos B = = . 2(4)(8) 64 Then, using the law of cosines on ABM, we get that √ √ 3 20 2 2 2 AM = 4 + 4 − 2(4)(4) cos B = √ ⇒ AD = . 3 2 Applying Power of a Point on M , √ √ 16 2 41 2 ( AM )( M E ) = ( BM )( M C ) ⇒ M E = ⇒ AE = . 3 6 Then, we note that AXB ∼ ACE , so that √ AC 60 2 AX 9 AX = AB ( ) = ⇒ = AE 41 AD 41 Guts Round A B X M C D E