HMMT 二月 1998 · 几何 · 第 10 题

1998 Harvard/MIT Math Tournament GEOMETRY Answer Sheet Name: School: Grade: 1 7 2 8 3 9 4 10a 5 10b 6 10c TOTAL:

GEOMETRY Question One . [3 points] Quadrilateral ALEX , pictured below (but not necessarily to scale!) can be inscribed in a circle; with m ∠ L A X = 20 ° and m ∠ A X E = 100 ° : L N E A X D Calculate m ∠ E D X . Question Two . [3 points] Anne and Lisa enter a park that has two concentric circular paths joined by two radial paths, one of which is at the point where they enter. Anne goes in to the inner circle along the first radial path, around by the shorter way to the second radial path and out exit along it to the exit. Walking at a the same rate, Lisa goes around the outer circle to the exit, taking the shorter enter of the two directions around the park. They arrive at the exit at the same time. The radial paths meet at the center of the park; what is the angle between them?

Question Three . [4 points] MD is a chord of length 2 in a circle of radius 1, and L is chosen on the circle so that the area of triangle MLD is the maximized. Find m ∠ M L D . Question Four . [4 points] A cube with side length 100cm is filled with water and has a hole through which the water drains into a cylinder of radius 100cm. If cm the water level in the cube is falling at a rate of 1 , how fast is s the water level in the cylinder rising? Question Five . [5 points] S Square SEAN has side length 2 and a N quarter-circle of radius 1 around E is cut out. Find the radius of the largest circle that can be inscribed in the remaining figure. A E Question Six . [5 points] A circle is inscribed in an equilateral triangle of side length 1. Tangents to the circle are drawn that cut off equilateral triangles at each corner. Circles are inscribed in each of these equilateral triangles. If this process is repeated infinitely many times, what is the sum of the areas of all the circles?

Question Seven . [5 points] Pyramid EARLY has rectangular base EARL and apex Y , and all of its edges are of integer length. The four edges from the apex have lengths 1, 4, 7, 8 (in no particular order), and EY is perpendicular to YR . Find the area of rectangle EARL . Question Eight . [6 points] It is not possible to construct a segment of length π using a straightedge, compass, and a given segment of length 1. The following construction, given in 1685 by Adam Kochansky, yields a segment whose length agrees with π to five decimal places: Construct a circle of radius 1 and call its center O . Construct a diameter AB of this circle and a line l tangent to the circle at A . Next, draw a circle with radius 1 centered at A , and call one of the intersections with the original circle C . Now from C draw an arc of radius 1 intersecting the circle around A at D , where D lies outside of the circle centered at O . Draw OD and let E be its point of intersection with l . Construct H on AE such that A is between H and E , and HE =3. The distance between B and H is then close to π ; calculate its exact value.

Question Nine . [7 points] Let T be the intersection of the common internal tangents of circles C , C with centers O , O respectively. Let P be one of the points 1 2 1 2 of tangency on C and let line l bisect angle O TP . Label the 1 1 intersection of l with C that is farthest from T , R , and label the 1 intersection of l with C that is closest to T , S . If C has radius 4, 2 1 C has radius 6, and O O = 20 , calculate ( TR )( TS ). 2 1 2 C 2 R P T S O O 2 1 C 1 l Question Ten [8 points]. Lukas is playing pool on a table shaped like an equilateral triangle. The pockets are at the corners of the triangle and are labeled C , H , and T . Each side of the table is 16 feet long. Lukas shoots a ball from corner C of the table in such a way that on the second bounce, the ball hits 2 feet away from him along side CH . a. [5 points] How many times will the ball bounce before hitting a pocket? b. [2 points] Which pocket will the ball hit? c. [1 points] How far will the ball travel before hitting the pocket?