HMMT 二月 2007 · 冲刺赛 · 第 18 题

HMMT February 2007 — Guts Round — Problem 18

[ 10 ] Convex quadrilateral ABCD has right angles ∠ A and ∠ C and is such that AB = BC and AD = CD . The diagonals AC and BD intersect at point M . Points P and Q lie on the circumcircle of triangle AM B and segment CD , respectively, such that points P, M , and Q are collinear. Suppose ◦ ◦ that m ∠ ABC = 160 and m ∠ QM C = 40 . Find M P · M Q , given that M C = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND √ 2 2 x +3 xy + y − 2 x − 2 y +4

解析

[ 10 ] Convex quadrilateral ABCD has right angles ∠ A and ∠ C and is such that AB = BC and AD = CD . The diagonals AC and BD intersect at point M . Points P and Q lie on the circumcircle of triangle AM B and segment CD , respectively, such that points P, M , and Q are collinear. Suppose ◦ ◦ that m ∠ ABC = 160 and m ∠ QM C = 40 . Find M P · M Q , given that M C = 6 . Answer: 36 . Note that m ∠ QP B = m ∠ M P B = m ∠ M AB = m ∠ CAB = ∠ BCA = ∠ CDB . Thus, M P · M Q = M B · M D . On the other hand, segment CM is an altitude of right triangle BCD , so 2 M B · M D = M C = 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND √ 2 2 x +3 xy + y − 2 x − 2 y +4