HMMT 二月 2014 · 团队赛 · 第 8 题

HMMT February 2014 — Team Round — Problem 8

[ 35 ] Let ABC be an acute triangle with circumcenter O such that AB = 4, AC = 5, and BC = 6. Let D be the foot of the altitude from A to BC , and E be the intersection of AO with BC . Suppose that X is on BC between D and E such that there is a point Y on AD satisfying XY k AO and Y O ? AX . Determine the length of BX .

解析

[ 35 ] Let ABC be an acute triangle with circumcenter O such that AB = 4, AC = 5, and BC = 6. Let D be the foot of the altitude from A to BC , and E be the intersection of AO with BC . Suppose that X is on BC between D and E such that there is a point Y on AD satisfying XY ‖ AO and Y O ⊥ AX . Determine the length of BX . Answer: 96/41 Let AX intersect the circumcircle of △ ABC again at K . Let OY intersect AK and BC at T and L , respectively. We have ∠ LOA = ∠ OY X = ∠ T DX = ∠ LAK , so AL is tangent to the circumcircle. Furthermore, OL ⊥ AK , so △ ALK is isosceles with AL = AK , so AK is also tangent to the circumcircle. Since BC and the tangents to the circumcircle at A and K all intersect at the same point L , CL is a symmedian of △ ACK . Then AK is a symmedian of △ ABC . Then we 2 ( AB ) BX 96 can use = to compute BX = . 2 XC ( AC ) 41