HMMT 二月 2007 · 冲刺赛 · 第 18 题
HMMT February 2007 — Guts Round — Problem 18
题目详情
英文原题
- [ 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 circumcircleof 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 .
10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUNDth
√
2 2
x +3 xy + y − 2 x − 2 y +4
解析
英文解析
- [ 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 circumcircleof 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
M B · M D = M C = 36.2
10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUNDth
√
2 2
x +3 xy + y − 2 x − 2 y +4