HMMT 十一月 2011 · GEN 赛 · 第 9 题

HMMT November 2011 — GEN Round — Problem 9

[ 7 ] Let P and Q be points on line l with P Q = 12. Two circles, ω and Ω, are both tangent to l at P and are externally tangent to each other. A line through Q intersects ω at A and B , with A closer to Q than B , such that AB = 10. Similarly, another line through Q intersects Ω at C and D , with C closer to Q than D , such that CD = 7. Find the ratio AD/BC . 7

解析

[ 7 ] Let P and Q be points on line l with P Q = 12. Two circles, ω and Ω, are both tangent to l at P and are externally tangent to each other. A line through Q intersects ω at A and B , with A closer to Q than B , such that AB = 10. Similarly, another line through Q intersects Ω at C and D , with C closer to Q than D , such that CD = 7. Find the ratio AD/BC . 8 2 Answer: We first apply the Power of a Point theorem repeatedly. Note that QA · QB = QP = 9 2 QC · QD . Substituting in our known values, we obtain QA ( QA + 10) = 12 = QC ( QC + 7). Solving these quadratics, we get that QA = 8 and QC = 9. AQ CQ We can see that = and that ∠ AQD = ∠ CQB , so QAD ∼ QCB . (Alternatively, going DQ BQ back to the equality QA · QB = QC · QD , we realize that this is just a Power of a Point theorem on the quadrilateral ABDC , and so this quadrilateral is cyclic. This implies that ∠ ADQ = ∠ ADC = AQ AD 8 ∠ ABC = ∠ QBC .) Thus, = = . BC QC 9 D 7 Ω C 9 Q P 12 ω 8 A 10 B 7