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2012哈佛大学-麻省理工数学竞赛(高中2月赛)geometry部分


GEOMETRY TEST
This test consists of 10 short-answer problems to be solved individually in 50 minutes. Problems will be weighted with point values after the contest based on how many comp

etitors solve each problem. There is no penalty for guessing. No translators, books, notes, slide rules, calculators, abaci, or other computational aids are permitted other than the of?cial translation sheets. Similarly, graph paper, rulers, protractors, compasses, and other drawing aids are not permitted. Our goal is that a closed form answer equivalent to the correct answer will be accepted. However, we do not always have the resourses to determine whether a complicated or strange answer is equivalent to ours. To assist us in awarding you all the points that you deserve, you answers should be simpli?ed as much as possible. Answers must be exact unless otherwise speci?ed. Correct mathematical notation must be used. No partial credit will be given unless otherwise speci?ed. If you believe the test contains an error, please submit your protest in writing to Science Center 109 during lunchtime. Enjoy!

15th Annual Harvard-MIT Mathematics Tournament
Saturday 11 February 2012

Geometry Test
1. ABC is an isosceles triangle with AB = 2 and ?ABC = 90? . D is the midpoint of BC and E is on AC such that the area of AEDB is twice the area of ECD. Find the length of DE. 2. Let ABC be a triangle with ∠A = 90? , AB = 1, and AC = 2. Let ? be a line through A perpendicular to BC, and let the perpendicular bisectors of AB and AC meet ? at E and F , respectively. Find the length of segment EF . 3. Let ABC be a triangle with incenter I. Let the circle centered at B and passing through I intersect side AB at D and let the circle centered at C passing through I intersect side AC at E. Suppose DE is the perpendicular bisector of AI. What are all possible measures of angle BAC in degrees? 4. There are circles ω1 and ω2 . They intersect in two points, one of which is the point A. B lies on ω1 such that AB is tangent to ω2 . The tangent to ω1 at B intersects ω2 at C and D, where D is the closer to B. AD intersects ω1 again at E. If BD = 3 and CD = 13, ?nd EB/ED. 5. A mouse lives in a circular cage with completely re?ective walls. At the edge of this cage, a small ?ashlight with vertex on the circle whose beam forms an angle of 15? is centered at an angle of 37.5? away from the center. The mouse will die in the dark. What fraction of the total area of the cage can keep the mouse alive?

37.5?
6. Triangle ABC is an equilateral triangle with side length 1. Let X0 , X1 , . . . be an in?nite sequence of points such that the following conditions hold: ? ? ? ? X0 is the center of ABC For all i ≥ 0, X2i+1 lies on segment AB and X2i+2 lies on segment AC. For all i ≥ 0, ?Xi Xi+1 Xi+2 = 90? . For all i ≥ 1, Xi+2 lies in triangle AXi Xi+1 .
∞ i=0

Find the maximum possible value of

|Xi Xi+1 |, where |P Q| is the length of line segment P Q.
2012 i=1

7. Let S be the set of the points (x1 , x2 , . . . , x2012 ) in 2012-dimensional space such that |x1 | + |x2 | + · · · + |x2012 | ≤ 1. Let T be the set of points in 2012-dimensional space such that max |xi | = 2. Let p be a randomly chosen point on T . What is the probability that the closest point in S to p is a vertex of S? 8. Hexagon ABCDEF has a circumscribed circle and an inscribed circle. If AB = 9, BC = 6, CD = 2, and EF = 4. Find {DE, F A}. 9. Let O, O1 , O2 , O3 , O4 be points such that O1 , O, O3 and O2 , O, O4 are collinear in that order, OO1 = √ 1, OO2 = 2, OO3 = 2, OO4 = 2, and ?O1 OO2 = 45? . Let ω1 , ω2 , ω3 , ω4 be the circles with respective centers O1 , O2 , O3 , O4 that go through O. Let A be the intersection of ω1 and ω2 , B be the intersection of ω2 and ω3 , C be the intersection of ω3 and ω4 , and D be the intersection of ω4 and ω1 , with A, B, C, D all distinct from O. What is the largest possible area of a convex quadrilateral P1 P2 P3 P4 such that Pi lies on Oi and that A, B, C, D all lie on its perimeter?

10. Let C denote the set of points (x, y) ∈ R2 such that x2 + y 2 ≤ 1. A sequence Ai = (xi , yi )|i ≥ 0 of points in R2 is ‘centric’ if it satis?es the following properties: ? A0 = (x0 , y0 ) = (0, 0), A1 = (x1 , y1 ) = (1, 0). ? For all n ≥ 0, the circumcenter of triangle An An+1 An+2 lies in C.
2 Let K be the maximum value of x2 2012 + y2012 over all centric sequences. Find all points (x, y) such 2 2 that x + y = K and there exists a centric sequence such that A2012 = (x, y).

15th Annual Harvard-MIT Mathematics Tournament
Saturday 11 February 2012

Geometry Test

Name School Team

Team ID#

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Score:


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