08amc12

USA AMC 12 2008 A 1 A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day’s job. At what time will the doughnut machine complete the job? (A) 1:50 PM (B) 3:00 PM (C) 3:30 PM (D) 4:30 PM (E) 5:50 PM 2 What is the reciprocal of 1 2 2 3 ? (A) 6 7 (B) 7 6 (C) 5 3 (D) 3 (E) 7 2 3 Suppose that 2 3 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much is 1 2 of 5 bananas? (A) 2 (B) 5 2 (C) 3 (D) 7 2 (E) 4 4 Which of the following is equal to the product 8 4 · 12 8 · 16 12 · · · 4n4 4n · · · 2008 2004 ? (A) 251 (B) 502 (C) 1004 (D) 2008 (E) 4016 5 Suppose that 2x 3 x 6 is an integer. Which of the following statements must be true about x? (A) It is negative. (B) It is even, but not necessarily a multiple of 3. (C) It is a multiple of 3, but not necessarily even. (D) It is a multiple of 6, but not necessarily a multiple of 12. (E) It is a multiple of 12. 6 Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a $90 rebate, and store B offers 25% off the same sticker price with no rebate. Heather saves $15 by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars? (A) 750 (B) 900 (C) 1000 (D) 1050 (E) 1500 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ USA AMC 12 2008 7 While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? (A) 2 (B) 4 (C) 6 (D) 8 (E) 10 8 What is the volume of a cube whose surface area is twice that of a cube with volume 1? (A) √ 2 (B) 2 (C) 2 √ 2 (D) 4 (E) 8 9 Older television screens have an aspect ratio of 4 : 3. That is, the ratio of the width to the height is 4 : 3. The aspect ratio of many movies is not 4 : 3, so they are sometimes shown on a television screen by ’letterboxing’ - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of 2 : 1 and is shown on an older television screen with a 27-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); filldraw((0,0)–(21.6,0)–(21.6,2.7)–(0,2.7)–cycle,grey,black); filldraw((0,13.5)– (21.6,13.5)–(21.6,16.2)–(0,16.2)–cycle,grey,black); draw((0,0)–(21.6,0)–(21.6,16.2)–(0,16.2)–cycle);[/asy] (A) 2 (B) 2.25 (C) 2.5 (D) 2.7 (E) 3 10 Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t? (A) ( 1 5 1 7 ) (t1)1 (B) ( 1 5 1 7 ) t11 (C) ( 1 5 1 7 ) t1 (D) ( 1 5 1 7 ) (t1)1 (E) (57)t1 11 Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the 13 visible numbers have the greatest possible sum. What is that sum? [asy]unitsize(.8cm); pen p = linewidth(1); draw(shift(-2,0)*unitsquare,p); label(quot;1quot;,(-1.5,0.5)); draw(shift(- 1,0)*unitsquare,p); label(quot;2quot;,(-0.5,0.5)); draw(unitsquare,p); label(quot;32quot;,(0.5,0.5)); draw(shift(1,0)*unitsquare,p); label(quot;16quot;,(1.5,0.5)); draw(shift(0,1)*unitsquare,p); la- bel(quot;4quot;,(0.5,1.5)); draw(shift(0,-1)*unitsquare,p); label(quot;8quot;,(0.5,-0.5));[/asy] (A) 154 (B) 159 (C) 164 (D) 167 (E) 189 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ USA AMC 12 2008 12 A function f has domain [0, 2] and range [0, 1]. (The notation [a, b] denotes {x : a ≤ x ≤ b}.) What are the domain and range, respectively, of the function g defined by g(x)1f(x1)? (A) [1, 1], [1, 0] (B) [1, 1], [0, 1] (C) [0, 2], [1, 0] (D) [1, 3], [1, 0] (E) [1, 3], [0, 1] 13 Points A and B lie ona circle centered at O, and ∠AOB60◦. A second circle is internally tangent to the first and tangent to both OA and OB. What is the ratio of the area of the smaller circle to that of the larger circle? (A) 1 16 (B) 1 9 (C) 1 8 (D) 1 6 (E) 1 4 14 What is the area of the region defined by the inequality |3x18||2y7| ≤ 3? (A) 3 (B) 7 2 (C) 4 (D) 9 2 (E) 5 15 Let k2008222008. What is the units digit of k22k? (A) 0 (B) 2 (C) 4 (D) 6 (E) 8 16 The numbers log(a3b7), log(a5b12), and log(a8b15) are the first three terms of an arithmetic sequence, and the 12th term of the sequence is log bn. What is n? (A) 40 (B) 56 (C) 76 (D) 112 (E) 143 17 Let a1, a2, . . . be a sequence of integers determined by the rule anan1/2 if an1 is even and an3an11 if an1 is odd. For how many positive integers a1 ≤ 2008 is it true that a1 is less than each of a2, a3, and a4? (A) 250 (B) 251 (C) 501 (D) 502 (E) 1004 18 Triangle ABC, with sides of length 5, 6, and 7, has one vertex on the positive x-axis, one on the positive y-axis, and one on the positive z-axis. Let O be the origin. What is the volume of tetrahedron OABC? (A) √ 85 (B) √ 90 (C) √ 95 (D) 10 (E) √ 105 19 In the expansion of( 1xx2 · · ·x27) (1xx2 · · ·x14)2, what is the coefficient of x28? (A) 195 (B) 196 (C) 224 (D) 378 (E) 405 20 Triangle ABC has AC3, BC4, and AB5. Point D is on AB, and CD bisects the right angle. The inscribed circles of 4ADC and 4BCD have radii ra and rb, respectively. What is ra/rb? http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 3 http://www.mathlinks.ro/ USA AMC 12 2008 (A) 1 28 ( 10 √ 2 ) (B) 3 56 ( 10 √ 2 ) (C) 1 14 ( 10 √ 2 ) (D) 5 56 ( 10 √ 2 ) (E) 3 28 ( 10 √ 2 ) 21 A permutation (a1, a2, a3, a4, a5) of (1, 2, 3, 4, 5) is heavy-tailed if a1a2 < a4a5. What is the number of heavy-tailed permutations? (A) 36 (B) 40 (C) 44 (D) 48 (E) 52 22 A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length x. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is x? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(- 2.687,-1.5513)–(-2.687,1.5513)–(-3.687,1.5513)–(-3.687,-1.5513)–cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label(quot;36;x36;quot;,(-2.687,0),E); label(quot;36;136;quot;,(-3.187,1.5513),S);[/asy] (A) 2 √ 5 √ 3 (B) 3 (C) 3 √ 7 √ 3 2 (D) 2 √ 3 (E) 52 √ 3 2 23 The solutions of the equation z44z3i6z24zii0 are the vertices of a convex polygon in the complex plane. What is the area of the polygon? (A) 25/8 (B) 23/4 (C) 2 (D) 25/4 (E) 23/2 24 Triangle ABC has ∠C60◦ and BC4. Point D is the midpoint of BC. What is the largest possible value of tan∠BAD? (A) √ 3 6 (B) √ 3 3 (C) √ 3 2 √ 2 (D) √ 3 4 √ 23 (E) 1 25 A sequence (a1, b1), (a2, b2), (a3, b3), . . . of points in the coordinate plane satisfies (an1, bn1)( √ 3anbn, √ 3bnan) for n1, 2, 3, . . .. Suppose that (a100, b100)(2, 4). What is a1b1? (A) minus 1 297 (B) minus 1 299 (C) 0 (D) 1 298 (E) 1 296 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 4 http://www.mathlinks.ro/ USA AMC 12 2008 B 1 A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 2 A 4 × 4 block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? 1 2 3 4 8 9 10 11 15 16 17 18 22 23 24 25 (A) 2 (B) 4 (C) 6 (D) 8 (E) 10 3 A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15, 000, and that the total of all players’ salaries for each team cannot exceed $700, 000. What is the maximum possiblle salary, in dollars, for a single player? (A) 270, 000 (B) 385, 000 (C) 400, 000 (D) 430, 000 (E) 700, 000 4 On circle O, points C and D are on the same side of diameter AB, ∠AOC30◦, and ∠DOB45◦. What is the ratio of the area of the smaller sector COD to the area of the circle? [asy]unitsize(6mm); defaultpen(linewidth(0.7)+fontsize(8pt)); pair C = 3*dir (30); pair D = 3*dir (135); pair A = 3*dir (0); pair B = 3*dir(180); pair O = (0,0); draw (Circle ((0, 0), 3)); label (quot;36;C36;quot;, C, NE); label (quot;36;D36;quot;, D, NW); label (quot;36;B36;quot;, B, W); label (quot;36;A36;quot;, A, E); label (quot;36;O36;quot;, O, S); label (quot;36;45◦36; quot; , (−0.3, 0.1),WNW ); label(quot; 36; 30◦36; quot; , (0.5, 0.1), ENE); draw(A− −B); draw(O −−D); draw(O −−C); [/asy] (A) 2 9 (B) 1 4 (C) 5 18 (D) 7 24 (E) 3 10 5 A class collects $50 to buy flowers for a classmate who is in the hospital. Roses cost $3 each, and carnations cost $2 each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50? (A) 1 (B) 7 (C) 9 (D) 16 (E) 17 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 5 http://www.mathlinks.ro/ USA AMC 12 2008 6 Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year? (A) 2500 (B) 3000 (C) 3500 (D) 4000 (E) 4500 7 For real numbers a and b, define a$b(ab)2. What is (xy)2$(yx)2? (A) 0 (B) x2y2 (C) 2x2 (D) 2y2 (E) 4xy 8 Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD? (A) 1 36 (B) 1 13 (C) 1 10 (D) 5 36 (E) 1 5 9 Points A and B are on a circle of radius 5 and AB6. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC? (A) √ 10 (B) 7 2 (C) √ 14 (D) √ 15 (E) 4 10 Bricklayer Brenda would take 9 hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together they talk a lot, and their combined output is decreased by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney? (A) 500 (B) 900 (C) 950 (D) 1000 (E) 1900 11 A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top 1 8 of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet? (A) 4000 (B) 2000(4 √ 2) (C) 6000 (D) 6400 (E) 7000 12 For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence? (A) 2008 (B) 4015 (C) 4016 (D) 4, 030, 056 (E) 4, 032, 064 13 Vertex E of equilateral 4ABE is in the interior of unit square ABCD. Let R be the region consisting of all points inside ABCD and outside 4ABE whose distance from AD is between 1 3 and 2 3 . What is the area of R? (A) 125 √ 3 72 (B) 125 √ 3 36 (C) √ 3 18 (D) 3 √ 3 9 (E) √ 3 12 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 6 http://www.mathlinks.ro/ USA AMC 12 2008 14 A circle has a radius of log10(a 2) and a circumference of log10(b 4). What is loga b? (A) 1 4pi (B) 1 pi (C) pi (D) 2pi (E) 102pi 15 On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let R be the region formed by the union of the square and all the triangles, and S be the smallest convex polygon that contains R. What is the area of the region that is inside S but outside R? (A) 1 4 (B) √ 2 4 (C) 1 (D) √ 3 (E) 2 √ 3 16 A rectangular floor measures a by b feet, where a and b are positive integers with b > a. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair (a, b)? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 17 Let A, B, and C be three distinct points on the graph of yx2 such that line AB is parallel to the x-axis and 4ABC is a right triangle with area 2008. What is the sum of the digits of the y-coordinate of C? (A) 16 (B) 17 (C) 18 (D) 19 (E) 20 18 A pyramid has a square base ABCD and vertex E. The area of square ABCD is 196, and the areas of 4ABE and 4CDE are 105 and 91, respectively. What is the volume of the pyramid? (A) 392 (B) 196 √ 6 (C) 392 √ 2 (D) 392 √ 3 (E) 784 19 A function f is defined by f(z)(4i)z2αzγ for all complex numbers z, where α and γ are complex numbers and i21. Suppose that f(1) and f(i) are both real. What is the smallest possible value of |α||γ| (A) 1 (B) √ 2 (C) 2 (D) 2 √ 2 (E) 4 20 Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200 feet along the path. A garbage truck travels at 10 feet per second in the same direction as Michael and stops for 30 seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 7 http://www.mathlinks.ro/ USA AMC 12 2008 21 Two circles of radius 1 are to be constructed as follows. The center of circle A is chosen uniformly and at random from the line segment joining (0, 0) and (2, 0). The center of circle B is chosen uniformly and at random, and independently of the first choice, from the line segment joining (0, 1) to (2, 1). What is the probability that circles A and B intersect? (A) 2 √ 2 4 (B) 3 √ 32 8 (C) 2 √ 21 2 (D) 2 √ 3 4 (E) 4 √ 33 4 22 A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? (A) 11 20 (B) 4 7 (C) 81 140 (D) 3 5 (E) 17 28 23 The sum of the base-10 logarithms of the divisors of 10n is 792. What is n? (A) 11 (B) 12 (C) 13 (D) 14 (E) 15 24 Let A0(0, 0). Distinct points A1, A2, . . . lie on the x-axis, and distinct points B1, B2, . . . lie on the graph of y √ x. For every positive integer n, An1BnAn is an equilateral triangle. What is the least n for which the length A0An ≥ 100? (A) 13 (B) 15 (C) 17 (D) 19 (E) 21 25 Let ABCD be a trapezoid with AB ‖ CD, AB11, BC5, CD19, and DA7. Bisectors of ∠A and ∠D meet at P , and bisectors of ∠B and ∠C meet at Q. What is the area of hexagon ABQCDP? (A) 28 √ 3 (B) 30 √ 3 (C) 32 √ 3 (D) 35 √ 3 (E) 36 √ 3 http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 8 http://www.mathlinks.ro/