SAT数学难题hardmathproblem.pdf

1. In the fi gure below, the graph of y = kx2intersects triangle ABC at B. If AB = BC and the area of trian- gle ABC is 6, what is the value of k? (3,0) B AC y = kx2 (1,0) 2. For all n, let n be defi ned as n= n2 n 2. If a and b are positive integers and a , b, which of the following could be true? I. (a + b)is even. II. (a + b)is odd. III. (a b)is negative. 3. The graphs of two linear functions, f and g, are per- pendicular. If f(2) = 4 and f(4) = 1, which of the following could defi ne g? (A) g(x) = 1 2 x + 4 (B) g(x) = x + 1 (C) g(x) = 2x + 4 (D) g(x) = 2x 3 (E) g(x) = x + 5 4. If (xh)(x+k) = x216, what is the value of h+k? (A) 8 (B) 4 (C)0 (D)4 (E)8 -3 -2 -11234567 -2 -1 1 2 3 4 5 6 7 5. A portion of the graph of the function h is shown in the xy-plane above. If h(4) h(s) = 4, which of the following could be the value of s? (A) 2 (B) 0 (C) 1 (D) 4 (E) 7 1 O P Q 6. In the fi gure above, three identical circles of radius 8 with centers O, P and Q are tangent to each other. What is the area of the shaded region bound by the circles? 7. The participants in a certain community service club come from all four grade levels in Smithville High School.There are equal numbers of sophomores and juniors. The number of seniors is twenty more than twice the number of sophomores and juniors combined, and there are 20 freshmen. If the number of sophomores,juniors and seniors combined accounts for 80 percent of the clubs participants, how many seniors are there in the club? 8. For integers a and b where a 0 and b 0, let the operation ? be defi ned as a ? b = ab a . Which of the following could be the value of a ? b? I. ab a1 II. 0 III. 1 9. The value of an investment increased by 20 percent to $800 by the end of the fi rst week. To the nearest dollar, what was the value of the investment at the beginning of the week? -3-2-11234567 -2 -1 1 2 3 4 5 6 7 y = f (x) y = g(x) 10. The fi gure above shows the graphs of the functions f and g. If g is defi ned in terms of a transformation of f such that g(x) = f(x+h)+k, what is the value of h k ? 11. For all x and y, let the operation x _ y be equal to the whole number remainder obtained when fi nding 2x y . If s is an even integer and the value of 7 _ s = 2, what is a possible value of s? 2 O P Q R S 30 12. In the fi gure above, OPQR is a square, and PS is the arc of a circle with center O. If the length of arc PS is , the area of the shaded sector is what fraction of the area of OPQR? (A) 2 (B) 4 (C) 8 (D) 12 (E) 15 13. Jack drove to work in the morning at an average speed of 45 miles per hour. He returned home in the evening along the same route and averaged 30 miles per hour.If Jack spent a total of one hour commuting to and from work, how many miles did he drive to work in the morning? 14. A student is weighing large and small marbles. While weighing one large marble with 8 small mar- bles, the student discovers that the large marbles weight is 5 times the average (arithmetic mean) of the weight of the small marbles. The large marbles weight is what fraction of the total weight of the 9 marbles? (1,0) AB C D E 15. In the fi gure above, rectangle ABCD lies on the coordinate plane with point D located at the origin. If AB = AE = ED, what is the area of quadrilateral BCDE? 105 150 A B C D E F Note: Figure not drawn to scale. 16. In the fi gure above, triangle ABC and parallelogram CDEF are constructed with B, C, and F along the same line. If the ratio of CF : BC = 1 : 2 and DE = 2 3, what is the area of triangle ABC? (A) 3 2 (B) 6 (C) 6 3 32 (D) 6 3 (E) 6 3 + 6 3 17. The function f is defi ned as f(x) = (x n)(x + m) for all values of x. If the function g is defi ned as g(x) = f(x) + 3, what is the y-intercept of the graph of g in terms of n and m? (A) 3n + 3m (B) 3n 3m (C) 3nm (D) 3 nm (E) nm 3 a, b, b a, a, . 18. In the sequence above, each term after the fi rst two is obtained by fi nding the diff erence between the two previous terms. If a = 3 and b = 4, what is the sum of the fi rst 200 terms of this sequence? g(x) = x2+ c h(x) = g(x 5) + 3 19. The equations of two functions, g and h, are shown above, where c is a constant integer. If the graph of h has two solutions, what is the maximum possible value of c? (A) 5 (B) 4 (C) 3 (D) 1 (E) 3 20. Triangle ABC has side lengths 3, 8, and a. Triangle DEF has side lengths 7, 10, and d. If a and d are both integers, what is the smallest possible value of (d a)? (A) 10 (B) 8 (C) 6 (D) 0 (E) 5 21. An antique coin collector has a number of coins and several display cases. If he puts one coin in each case, there are four coins left out of the cases. If instead he puts three coins in each case, all coins are placed, and there are six empty cases left over. How many coins does the collector have? x A B Start Stop 22. A small circle, A, is rolled around the circumference of a larger circle, B, as shown in the fi gure above. Circle A starts at the position indicated “Start” and makes two complete rotations, coming to rest at the position marked “Stop”. If the ratio of the areas of A to B is 1 to 25, what is the value of x? 4 23. When the positive integer w is divided by 6, the remainder is 3.When the positive integer x is divided by 7, the remainder is 5. How many possible values of wx are there such that wx 0 y 7 40. If the statements above are true and y is a prime integer, what is the smallest possible integer value of x? 41. The quadratic function h is given by h(x) = ax2, where a is a positive constant. If h(m) = h(n) for somevaluesofmandn, whichofthefollowingcould be true? I. m = n II. m = n III. h(m) = h(n) (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 42. What is the maximum number of intersection points of a circle with radius r and a square with side length 2r? (A) 2 (B) 4 (C) 5 (D) 6 (E) 8 43. If 4s6w = 2t +18 and 3w2s = 5+t, what is the value of t? (A) 7 (B) 3 (C)0 (D)4 (E) 10 44. The average of a set of fi ve positive integers is a. When the number 9 is added to the set, the average of the numbers increases by 1. What is the value of a? (A) 12 (B)9 (C)8 (D)3 (E)1 45. A cylindrical tank with a base radius of 3 feet and height of 12 feet is fi lled with water to two-thirds of its capacity. The water is then transferred to a larger cylindrical container with a base radius of 2 feet. What is the height of the water, in feet, in the the larger container? 8 r t Note: Figure not drawn to scale. 46. In the fi gure above, a circular track is shown. The outside of the track is a distance r meters from the center, and the inside of the track is t meters from the center. If the area of the shaded portion of the track shown is 28 square meters and t = 3 4r, what is the value of t? 47. Of 5 members of a high school club, 3 are to be as- signed to the fi nance committee and 2 are to be as- signed to the activities committee. If 3 of the mem- bers are seniors and 2 are juniors, and if those as- signed to the fi nance committee are to be chosen at random, what is the probability that the fi nance com- mittee assignments will be given to 2 of the seniors and 1 of the juniors? (A) 1 3 (B) 2 5 (C) 1 2 (D) 3 5 (E) 2 3 48. If x 2 3= y6and x8= yt, what is the value of t? 49. To make a cran-apple juice, 3 parts of cranberry juice are mixed with 2 parts of apple juice.To make an apple-pomegranate juice, 2 parts of pomegranate juice are mixed with 1 part of apple juice. If equal amounts of cran-apple and apple-pomegranate juice are mixed, what fraction of the new mixture is apple juice? (A) 3 16 (B) 1 4 (C) 11 30 (D) 3 8 (E) 7 12 50. For a poll, 72 people were asked about two models of a car, A and B. Of the people polled, 38 have owned model A, 42 have owned model B, and 10 have owned neither model. How many of the people polled have owned both models? 51. In choosing a color scheme for her living room, Julia must pick 2 colors from a list of 6 available paint colors. How many diff erent two-color schemes are available for Julia to use for painting her living room? 9 O P Q 52. In the fi gure above, the circle has center O and radius r, and triangle OPQ is equilateral. If the area of the shaded region is 120 and PQ = 3 2r, what is the area of triangle OPQ? 53. There are blue and red shirts on a clothing display. If 15 of the shirts are blue and x 7 of the shirts are red where x is an integer, what is the minimum number of shirts that could be on display? (A) 16 (B) 18 (C) 19 (D) 20 (E) 21 P 20 m x Note: Figure not drawn to scale. 54. In the fi gure above, line m is tangent to a vertex of a regular octagon at point P. What is the value of x? 55. If the function g is defi ned as g(x) = 2x2 3x and g(t + 1) = 0, what is the value of g(t + 1.5) if t 0? 56. If a is inversely proportional to the square of b, and a = 4 when b = 8, what is the value of b when a = 16? (A) 2 (B) 4 (C) 8 (D) 16 (E) 32 57. If x, y, and z are positive integers and xz yz= 225, what is the value of xyz? (A) 90 (B) 60 (C) 45 (D) 30 (E) 15 10 A B C 58. The cube above has a side length of x. Point A is at the center of the top face of the cube, and points B andC are midpoints of the bottom edges of the cube. In terms of x, what is the perimeter of 4ABC? (A) x + x 2 (B) x + x 3 (C) x + x 5 (D) x + 2x 2 (E) 3x 120 A B O 4 59. In the fi gure above, 4ABO is inscribed in the circle with center O.If arc AB measures 4 and AOB = 120, what is the length of AB? 60. Some of Williams friends arranged to contribute $6 each to buy him a birthday cake. Two of the friends then decided to use their money to buy a diff erent present, and the remaining friends had to contribute an extra $3 each to make up the diff erence for the cost of the cake. How many friends were originally going to contribute to the cost of the cake? 61. On his fi rst day reading a book, Henrik reads 1 4 of the books pages. On the second day, he reads 1 4 of the pages that remain, leaving 189 pages left to read. What is the total number of pages in Henriks book? 62. On the xy-coordinate plane, lines and m are perpendicular and intersect at (2,0). If b is the y- intercept of line and c is the y-intercept of line m, what is the value of bc? (A)4 (B)2 (C)1 (D) 1 (E) 4 2,4,8,16,. 63. In the sequence above, every term after the fi rst is obtained by multiplying the preceding term by 2. The sum of which pair of terms in the sequence is equal to 249? (A) 24thand 25th (B) 25thand 26th (C) 48thand 49th (D) 49thand 50th (E) 50thand 51st 11 A B C D E F 64. A semicircular window is divided into four equal panes, as shown in the fi gure above. A straight line drawn between points B and D (not shown) is 16 feet long. What is the perimeter of one of the panes? (A) 16 (B) 32 (C) (16 + 2) 2 (D) 48 (E) (32 + 2) 2 65. When the square of a number is raised to 10 times the number, the result is equal to 4 raised to 5 times the number. What is the square of the number? P (3, 2) O Q (0, b) 66. In the fi gure above, 4OPQ is shown on the xy-plane. What is the value of b? xy 6427 16216 67. Given the table of x and y values above, y could be inversely proportional to which of the following? (A) x (B) x (C) x 2 3 (D) x 3 2 (E) 2 3x 68. For all numbers x, y, and z, let ?(x,y,z) be defi ned as ?(x,y,z) = xy + yz + x + z. For all numbers a, b, and c where b = 1 and a c 0, for how many ordered pairs (a,c) will ?(a,b,c) equal zero? (A) None (B) One (C) Two (D) Three (E) More than three 150 x y A B C D m n 69. In the fi gure above, mkn, and 4ABC has two side lengths x and y. If AB bisects DBC, what is the ratio of x y ? (A) 2 : 2 (B) 2 : 1 (C) 3 : 2 (D) 3 : 1 (E)2 : 1 12 70. For all integers n and m, let n ? m be defi ned as the sum of all integers from n to m, inclusive. What is the value of (2 ? 100) (1 ? 99)? A B C E D 71. In the fi gure above, square ABDE is inscribed in a circle, and 4ACE has vertices on the square. What is the ratio of the area of the circle to the area of triangle ABDE? (A) 4 3 (B) (C) 2 3 (D) 2 (E) 4 72. On a high school track team, there are initially 60 students, and 35 percent of the team members are girls. After several more girls join the team, the per- centage of girls on the team increases to 40 percent. The number of boys on the team is how much greater than the number of girls? (A) 5 (B) 7 (C) 11 (D) 13 (E) 18 73. In the fi gure above, there are four squares each with a perimeter of 8. The four points indicated are at the centers of their respective squares. What is the area of the square formed by connecting the four points (not shown)? (A) 4 (B) 8 (C) 16 (D) 20 (E) 32 A B C D E m n 74. In the fi gure above, m k n, AB = DE = 2, and CD = 3. What is the length of BD? 13 k2(a b)2= 3a2 6ab + 3b2 75. The equation above is true for all values of a and b. If k is a constant greater than zero, what is the value of k? (A) 0 (B) 3 (C) 6 (D) 3 (E) 6 O P Q R S x 76. In the fi gure above, QR and PS are arcs of circles with the same center O. If segment OS is 3 5 of OR, PQ = 8, and 5.0ptPS _ = 4 3, what is the value of x? 77. Four friends are arranged on a set of steps. Samuel stands on the middle step, Trisha stands 5 steps above him, David is 2 steps below Trisha, and Martin is 9 steps above David. How many steps are there in total? 78. If 4x3+ 2x2= 16x + 8, and x is a non-zero integer, what is the value of x2? O P k 79. The fi gure above shows the graph of a circle with center O tangent to the x-axis at point P. Line k passes through the origin and the center of the cir- cle. If the radius of the circle has length 2 and the area of the shaded region is 3 , what is the slope of line k? (A) 1 2 (B) 1 2 (C) 1 3 (D) 1 3 (E) 1 4 PRICES AT ABBEYS SANDWICH SHOP ORDERPRICE Sandwich Salad$16.00 Salad Soup$10.00 Sandwich Soup$18.00 80. The table above shows the prices of lunch combina- tion orders at Abbeys Sandwich Shop for the month of May. The orders include diff erent combinations of three items: a sandwich, a soup, and a salad. What is the average price, in dollars, of these items? 14 81. Five guitars, labeled V, W, X, Y, and Z, are to be arranged in a line in a display case. Guitar X must be positioned furthest to the right, and guitars V and Y must be positioned next to one another. What fraction of the total possible arrangements will result in guitars W and Z being positioned next to one another? r O 82. In the fi gure above, a small circle with radius r is inscribed in a square (shaded), and the square is in- scribed in a larger circle. What is the ratio of the area of the larger circle to the area of the smaller circle? (A) 2 : 1 (B) 2 2 : 1 (C) 3 : 1 (D) 3 2 : 1 (E) 4 : 1 83. If f and g are functions such that f(a) = 2g(2a) and g(m) = 6m + 10, what is the value of f(5) ? (A) 10 (B) 20 (C) 100 (D) 130 (E) 140 A B y = 2x3! y = 2x! 84. In the fi gure above, the graphs of y = 2x3and y = 2x intersect at points A and B. If the diameter of a circle (not shown) is formed by AB, what is the area of the circle? (A) 8 (B) 6 (C) 5 (D) 4 (E) 2 TRANSMISSION REPAIRS! 42%! BRAKE TUNE-UPS! 22%! INSPECTIONS! 6%! OIL CHANGES! 19%! TIRE ROTATIONS! 11%! PETES AUTOMOTIVE REVENUES! 85. The circle graph above shows the revenues for Petes Automotive last year. During the year, transmissions and inspections revenues totaled $240,000.How much more in revenue did Petes Automotive earn through oil changes than through tire rotations? (A) $8,000 (B) $11,000 (C) $19,000 (D) $40,000 (E) $260,000 15 A B C D E F G a b c d f g 86. In the fi gure above, DEF is a right angle. What is the average of a,b,c,d, f, and g? 87. A positive number greater than 1 is decreased by p percent. The resulting number must be increased by n percent to obtain the original value. If n and p are positive numbers such that n 100 and p 100, which of the following expresses n as a percent in terms of p? (A) 100 p p % (B) p 100 p % (C) p 100(p 1) % (D) 100(p 1) 100 p % (E) 100p 100 p % 88. There are 42 marbles in a bag, and each marble is colored one of three colors. If more than one-third of the marbles are blue, what is the greatest possible fraction of marbles in the bag that could be one of the other colors? (A) 2 3 (B) 9 14 (C) 1