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AMC12历年试卷


USA
AMC 12 2000

1 In the year 2001, the United States will host the International Mathematical Olympiad. Let I, M, and O be distinct positive integers such that the product I

· M · O = 2001. What’s the largest possible value of the sum I + M + O? (A)23 (B)55 (C)99 (B)40002000 (D)111 (E)671 (D)4, 000, 0002000 (E)20004,000,000 2 2000(20002000 ) = (A)20002001 (C)20004000 3 Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally? (A)40 (B)50 (C)55 (D)60 (E)75 4 The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence? (A)0 5 If |x (A) 2 (B)4 (B)2 (C)6 (C)2 (D)7 2p (E)9 p= (D)2p 2 (E)|2p 2| 2| = p, where x < 2, then x

6 Two di?erent prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? (A)21 (A)0 (B)60 (B)1 (C)119 (C)2 (D)180 (D)3 (E)4 (E)231 7 How many positive integers b have the property that logb 729 is a positive integer? 8 Figures 0, 1, 2, and 3 consist of 1, 5, 13, and 25 non-overlapping squares. If the pattern continued, how many non-overlapping squares would there be in ?gure 100? (A)10401 (A) 71 (B) 76 (B)19801 (C) 80 (C)20201 (D) 82 (D)39801 (E) 91 (E)40801

[img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 522sid = 05be4d7cd9bf b043e26947 10 9 The point P = (1, 2, 3) is re?ected in the xy -plane, then its image Q is rotated by 180 about the x-axis to produce R, and ?nally, R is translated by 5 units in the positive-y direction to produce S . What are the coordinates of S ? (A) (1, 7, 3) (B) ( 1, 7, 3) (C) ( 1, 2, 8) (D) ( 1, 3, 3) (E) (1, 3, 3)

USA
AMC 12 2000

11 Two non-zero real numbers, a and b, satisfy ab = a a b of + ab? b a 1 1 1 (A) 2 (B) (C) (D) (E) 2 2 3 2

b. Which of the following is a possible value

12 Let A, M , and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A · M · C + A · M + M · C + C · A? (A) 62 (B) 72 (C) 92 (D) 102 (E) 112 13 One morning each member of Angelas family drank an 8-ounce mixture of coee with milk. The amounts of coee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coee. How many people are in the family? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 14 When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of x? (A) 3 (B) 6 (C) 9 (D) 17 (E) 20 15 Let f be a function for which f (x/3) = x2 + x + 1. Find the sum of all values of z for which f (3z ) = 7. (A) 1/3 (B) 1/9 (C) 0 (D) 5/9 (E) 5/3 16 A checkerboard of 13 rows and 17 columns has a number written in each square, beginning in the upper left corner, so that the rst row is numbered 1, 2, . . . , 17, the second row 18, 19, . . . , 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, . . . , 13, the second column 14, 15, . . . , 26 and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system). (A) 222 (B) 333 (C) 444 (D) 555 (E) 666 17 A circle centered at O has radius 1 and contains the point A. Segment AB is tangent to the circle at A and \AOB = ?. If point C lies on OA and BC bisects \ABO, then OC = [asy]import olympiad; unitsize(1cm); defaultpen(fontsize(10pt)); labelmargin=0.2; dotfactor=4; pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O); pair C=extension(B,D,O,A); draw(Circle(O,1)); draw(O–A–B–cycle); draw(B–C);

USA
AMC 12 2000

label(quot;36;O36;quot;,O,SW); dot(O); label(quot;36;?36; quot; , (0.1, 0.05), EN E ); dot(C); label(quot;36;C36;quot;,C,S); dot(A); label(quot;36;A36;quot;,A,E); dot(B); label(quot;36;B36;quot;,B,E);[/asy] (A) sec2 ? tan ? (B) 1 2 (C) cos2 ? 1 + sin ? (D) 1 1 + sin ? (E) sin ? cos2 ?

18 In year N , the 300th day of the year is a Tuesday. In year N + 1, the 200th day of the year is also a Tuesday. On what day of the week did the 100th day of year N 1 occur? (A) Thursday (B) Friday (C) Saturday (D) Sunday (E) Monday 19 In triangle ABC , AB = 13, BC = 14, and AC = 15. Let D denote the midpoint of BC and let E denote the intersection of BC with the bisector of angle BAC . Which of the following is closest to the area of the triangle ADE ? (A) 2 (B) 2.5 (C) 3 (D) 3.5 (E) 4 20 If x, y , and z are positive numbers satisfying x + 1/y = 4, y + 1/z = 1, and z + 1/x = 7/3 then xyz = (A) 2/3 (B) 1 (C) 4/3 (D) 2 (E) 7/3 21 Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is m times the area of the square. The ratio of the area of the other small right triangle to the area of the square is 1 1 1 (A) (B) m (C) 1 m (D) (E) 2m + 1 4m 8m2 22 The graph below shows a portion of the curve de?ned by the quartic polynomial P (x) = x4 + ax3 + bx2 + cx + d. Which of the following is the smallest? (A) P ( 1) (B) The product of the zeros of P (C) The product of the non - real zeros of P (D) The sum of the coe cients of P (E) The sum of the real zeros of P [img]13164[/img]

This ?le was downloaded from the AoPS http://www.artofproblemsolving.com/

MathLinks Math Olympiad Resources Page Page 3 http://www.mathlinks.ro/

USA
AMC 12 2000

23 Professor Gamble buys a lottery ticket, which requires that he pick six dierent integers from 1 through 46, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? (A) 1/5
_

(B) 1/4
_

(C) 1/3

(D) 1/2

(E) 1
_

24 If circular arcs AC and BC have centers at B and A, respectively, then there exists a circle tangent to both AC and BC , and to AB . If the length of BC is 12, then the circumference of the circle is [asy]unitsize(2cm); defaultpen(fontsize(8pt));

pair O=(0,.375); pair A=(-.5,0); pair B=(.5,0); pair C=shift(-.5,0)*dir(60); draw(Arc(A,1,0,60)); draw(Arc(B,1,120,180)); draw(A–B); draw(Circle(O,.375)); dot(A); dot(B); dot(C); label(quot;36;A36;quot;,A,SW) label(quot;36;B36;quot;,B,SE); label(quot;36;C36;quot;,C,N);[/asy] (A) 24 (B) 25 (C) 26 (D) 27 (E) 28 25 Eight congruent equilateral triangles, each of a dierent color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; unitsize(1.5cm); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A–B–E–cycle); draw(A–C–D–cycle); draw(F–C–B–cycle); draw(F–D–E–cycle,dotted+linewidth(0.7));[/asy] (A) 210 (B) 560 (C) 840 (D) 1260 (E) 1680

USA
AMC 12 2001

1 The sum of two numbers is S . Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the ?nal two numbers? (A) 2S + 3 (B) 3S + 2 (C) 3S + 6 (D) 2S + 6 (E) 2S + 12 2 Let P (n) and S (n) denote the product and the sum, respectively, of the digits of the integer n. For example, P (23) = 6 and S (23) = 5. Suppose N is a two-digit number such that N = P (N ) + S (N ). What is the units digit of N ? (A) 2 (B) 3 (C) 6 (D) 8 (E) 9 3 The state income tax where Kristin lives is levied at the rate of p% of the ?rst U N IQaf a0a9a2b QIN U 28000 of annual income plus (p + 2)% of any amount above U N IQaf a0a9a2b QIN U 28000. Kristin noticed that the state income tax she paid amounted to (p + 0.25)% of her annual income. What was her annual income? (A) U N IQaf a0a9a2b QIN U 28000 (B) U N IQaf a0a9a2b QIN U 32000 (C) U N IQaf a0a9a2b QIN U 35000 (D) U N IQaf a0a9a2b QIN U 42000 (E) U N IQaf a0a9a2b QIN U 56000 4 The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum? (A) 5 (B) 20 (C) 25 10000! 25000 (D) 30 9999! 25000 (E) 36 10000! 25000 · 5000! 5000! 25000 5 What is the product of all odd positive integers less than 10000? (A) 10000! (5000!)2 (B) (C) (D) (E)

6 A telephone number has the form ABC DEF GHIJ , where each letter represents a di?erent digit. The digits in each part of the numbers are in decreasing order; that is, A > B > C , D > E > F , and G > H > I > J . Furthermore, D, E , and F are consecutive even digits; G, H , I , and J are consecutive odd digits; and A + B + C = 9. Find A. (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 7 A charity sells 140 bene?t tickets for a total of U N IQ975f 5815e QIN U 2001. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? (A) U N IQ975f 5815e QIN U 782 (B) U N IQ975f 5815e QIN U 986 (C) U N IQ975f 5815e QIN U 1158 (D) U N IQ975f 5815e QIN U 1219 (E) U N IQ975f 5815e QIN U 1449 8 Which of the cones listed below can be formed from a 252 sector of a circle of radius 10 by aligning the two straight sides? [asy]import graph; unitsize(1.5cm); defaultpen(fontsize(8pt));

USA
AMC 12 2001

draw(Arc((0,0),1,-72,180),linewidth(.8pt)); draw(dir(288)–(0,0)–(-1,0),linewidth(.8pt)); label(quot;36;1036;qu 0.5,0),S); draw(Arc((0,0),0.1,-72,180)); label(quot;36;252 36; quot; , (0.05, 0.05), N E ); [/asy ](A) A cone with slant height of 10 and radius 6 (B) A cone with height of 10 and radius 6 (C) A cone with slant height of 10 and radius 7 (D) A cone with height of 10 and radius 7 (E) A cone with slant height of 10 and radius 8 9 Let f be a function satisfying f (xy ) = f (x)/y for all positive real numbers x and y . If f (500) = 3, what is the value of f (600)? 5 18 (A) 1 (B) 2 (C) (D) 3 (E) 2 5 10 The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to (A) 50 (B) 52 (C) 54 (D) 56 (E) 58 [asy]unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)–(3,0)–(3,3)–(0,3)–(0,0); path p2=(0,1)–(1,1)–(1,0); path p3=(2,0)–(2,1)–(3,1); path p4=(3,2)–(2,2)–(2,3); path p5=(1,3)–(1,2)–(0,2); path p6=(1,1)–(2,2); path p7=(2,1)– (1,2); path[] p=p1020304050607; for(int i=0; i?3; ++i) for(int j=0; j?3; ++j) draw(shift(3*i,3*j)*p); [/asy] 11 A box contains exactly ?ve chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? 3 2 1 3 7 (A) (B) (C) (D) (E) 10 5 2 5 10 12 How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5? (A) 768 (B) 801 (C) 934 (D) 1067 (E) 1167 13 The parabola with equation y = ax2 + bx + c and vertex (h, k ) is re?ected about the line y = k . This results in the parabola with equation y = dx2 + ex + f . Which of the following equals a + b + c + d + e + f ? (A) 2b (B) 2c (C) 2a + 2b (D) 2h (E) 2k 14 Given the nine-sided regular polygon A1 A2 A3 A4 A5 A6 A7 A8 A9 , how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set {A1 , A2 , ...A9 }? (A) 30 (B) 36 (C) 63 (D) 66 (E) 72 15 An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of

USA
AMC 12 2001

the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) p 1p 3 (A) 3 (B) 1 (C) 2 (D) (E) 2 2 2 16 A spider has one sock and one shoe for each of its eight legs. In how many di?erent orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? 16! (A) 8! (B) 28 · 8! (C) (8!)2 (D) 8 (E) 16! 2 17 A point P is selected at random from the interior of the pentagon with vertices A = (0, 2), B = (4, 0), C = (2? + 1, 0), D = (2? + 1, 4), and E = (0, 4). What is the probability that \AP B is obtuse? 1 1 5 3 1 (A) (B) (C) (D) (E) 5 4 16 8 2 18 A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the ?rst two and to one of their common external tangents as shown. The radius of the third circle is 1 2 5 4 1 (A) (B) (C) (D) (E) 3 5 12 9 2 19 The polynomial P (x) = x3 + ax2 + bx + c has the property that the mean of its zeros, the product of its zeros, and the sum of its coe cients are all equal. If the y -intercept of the graph of y = P (x) is 2, what is b? (A) 11 (B) 10 (C) 9 (D) 1 (E) 5 20 Points A = (3, 9), B = (1, 1), C = (5, 3), and D = (a, b) lie in the ?rst quadrant and are the vertices of quadrilateral ABCD. The quadrilateral formed by joining the midpoints of AB, BC, CD, and DA is a square. What is the sum of the coordinates of point D? (A) 7 (B) 9 (C) 10 (D) 12 (E) 16 21 Four positive integers a, b, c, and d have a product of 8! and satisfy ab + a + b = 524bc + b + c = 146cd + c + d = 104 What is a (A) 4 d? (B) 6 (C) 8 (D) 10 (E) 12

USA
AMC 12 2001

22 In rectangle ABCD, points F and G lie on AB so that AF = F G = GB and E is the midpoint of DC . Also, AC intersects EF at H and EG at J . The area of the rectangle ABCD is 70. Find the area of triangle EHJ . 5 35 7 35 (A) (B) (C) 3 (D) (E) 2 12 2 8 23 A polynomial of degree four with leading coe cient 1 and integer coe cients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? p p 1 + i 11 1+i 1 i 1 + i 13 (A) (B) (C) + i (D) 1 + (E) 2 2 2 2 2 24 In 4ABC , \ABC = 45 . Point D is on BC so that 2 · BD = CD and \DAB = 15 . Find \ACB . (A) 54 (B) 60 (C) 72 (D) 75 (E) 90 25 Consider sequences of positive real numbers of the form x, 2000, y, ..., in which every term after the ?rst is 1 less than the product of its two immediate neighbors. For how many di?erent values of x does the term 2001 appear somewhere in the sequence? (A) 1 (B) 2 (C) 3 (D) 4 (E) more than 4

USA
AMC 12 2002

A

1 Compute the sum of all the roots of (2x + 3)(x (A) 7/2 (B) 4 (C) 5 (D) 7

4) + (2x + 3)(x

6) = 0.

(E) 13

2 Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? (A) 15
? 2(

(B) 34 )
?

(C) 43

(D) 51

(E) 138

3 According to the standard convention for exponentiation, 2
2 22 22

=2

= 216 = 65, 536.

If the order in which the exponentiations are performed is changed, how many other values are possible? (A) 0 (A) 48 (B) 1 (B) 60 (C) 2 (C) 75 (D) 3 (E) 4 (E) 150 4 Find the degree measure of an angle whose complement is 25% of its supplement. (D) 120 5 Each of the small circles in the ?gure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

[asy]unitsize(.3cm); path c=Circle((0,2),1); ?lldraw(Circle((0,0),3),grey,black); ?lldraw(Circle((0,0),1),white,b ?lldraw(c,white,black); ?lldraw(rotate(60)*c,white,black); ?lldraw(rotate(120)*c,white,black); ?lldraw(rotate(180)*c,white,black); ?lldraw(rotate(240)*c,white,black); ?lldraw(rotate(300)*c,white,black);[/ (A) ? (B) 1.5? (C) 2? (D) 3? (E) 3.5? 6 For how many positive integers m does there exist at least one positive integer n such that m · n ? m + n? (A) 4 (B) 6 (C) 9 (D) 12 (E) in?nitely many 7 If an arc of 45 on circle A has the same length as an arc of 30 on circle B , then the ratio of the area of circle A to the area of circle B is 4 2 5 3 9 (A) (B) (C) (D) (E) 9 3 6 2 4

USA
AMC 12 2002

8 Betsy designed a ?ag using blue triangles, small white squares, and a red center square, as shown. Let B be the total area of the blue triangles, W the total area of the white squares, and R the area of the red square. Which of the following is correct?

[asy]unitsize(3mm); ?ll((-4,-4)–(-4,4)–(4,4)–(4,-4)–cycle,blue); ?ll((-2,-2)–(-2,2)–(2,2)–(2,-2)– cycle,red); path onewhite=(-3,3)–(-2,4)–(-1,3)–(-2,2)–(-3,3)–(-1,3)–(0,4)–(1,3)–(0,2)–(-1,3)–(1,3)– (2,4)–(3,3)–(2,2)–(1,3)–cycle; path divider=(-2,2)–(-3,3)–cycle; ?ll(onewhite,white); ?ll(rotate(90)*onewhite,w ?ll(rotate(180)*onewhite,white); ?ll(rotate(270)*onewhite,white);[/asy] (A) B = W (B) W = R (C) B = R (D) 3B = 2R (E) 2R = W 9 Jamal wants to store 30 computer ?les on ?oppy disks, each of which has a capacity of 1.44 megabytes (MB). Three of his ?les require 0.8 MB of memory each, 12 more require 0.7 MB each, and the remaining 15 require 0.4 MB each. No ?le can be split between ?oppy disks. What is the minimal number of ?oppy disks that will hold all the ?les? (A) 12 (B) 13 (C) 14 (D) 15 (E) 16 10 Sarah pours four ounces of co?ee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the co?ee from the ?rst cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the ?rst. What fraction of the liquid in the ?rst cup is now cream? (A) 1/4 (B) 1/3 (C) 3/8 (D) 2/5 (E) 1/2 11 Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? (A) 45 (B) 48 (C) 50 (D) 55 (E) 58 63x + k = 0 are prime numbers. The number of (E) more than four 12 Both roots of the quadratic equation x2 possible values of k is (A) 0 (B) 1 (C) 2 (D) 3

13 Two di?erent numbers a and b each di?er from their reciprocals by 1. What is a + b? p p (A) 1 (B) 2 (C) 5 (D) 6 (E) 3 14 For all positive integers n, let f (n) = log2002 n2 . Let N = f (11) + f (13) + f (14) Which of the following relations is true? (A) N < 1 (B) N = 1 (C) 1 < N < 2 (D) N = 2 (E) N > 2

USA
AMC 12 2002

15 The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is (A) 11 (B) 12 (C) 13 (D) 14 (E) 15 16 Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5} and Sergio randomly selects a number from the set {1, 2, ..., 10}. The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is (A) 2/5 (B) 9/20 (C) 1/2 (D) 11/20 (E) 24/25 17 Several sets of prime numbers, such as {7, 83, 421, 659} use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? (A) 193 (x and (x + 15)2 + y 2 = 81, respectively. What is the length of the shortest line segment P Q that is tangent to C1 at P and to C2 at Q? (A) 15 (B) 18 (C) 20 (D) 21 (E) 24 19 The graph of the function f is shown below. How many solutions does the equation f (f (x)) = 6 have? [img]13247[/img] (A) 2 (B) 4 (C) 5 (D) 6 (E) 7 20 Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.ab is expressed as a fraction in lowest terms. How many di?erent denominators are possible? (A) 3 (B) 4 (C) 5 (D) 8 (E) 9 21 Consider the sequence of numbers: 4, 7, 1, 8, 9, 7, 6, . . . . For n > 2, the nth term of the sequence is the units digit of the sum of the two previous terms. Let Sn denote the sum of the ?rst n terms of this sequence. The smallest value of n for which Sn > 10, 000 is: (A) 1992 (B) 1999 (C) 2001 (D) 2002 (E) 2004 22 Triangle ABC is a right triangle with \ACB as its right angle, m\ABC = 60 , and AB = 10. Let P be randomly chosen p inside 4ABC , and extend BP to meet AC at D. What is the probability that BD > 5 2? (B) 207 (C) 225 (D) 252 (E) 447 18 Let C1 and C2 be circles de?ned by 10)2 + y 2 = 36

USA
AMC 12 2002

[asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A–C–B–cycle); draw(B–D); dot(P); label(quot;Aquot;,A,S); label(quot;Dquot;,D,S); label(quot;Cquot;,C,S); label(quot;Pquot;,P,NE); label(quot;Bquot;,B,N);[/asy] p p p 2 2 3 5 1 3 1 5 (A) (B) (C) (D) (E) 2 3 3 2 5 23 In triangle ABC , side AC and the perpendicular bisector of BC meet in point D, and BD bisects \ABC . If AD = 9 and DC = 7, what is the area of triangle ABD? p p (A) 14 (B) 21 (C) 28 (D) 14 5 (E) 28 5 24 Find the number of ordered pairs of real numbers (a, b) such that (a + bi)2002 = a (A) 1001 (B) 1002 (C) 2001 (D) 2002 (E) 2004 bi.

25 The nonzero coe cients of a polynomial P with real coe cients are all replaced by their mean to form a polynomial Q. Which of the following could be a graph of y = P (x) and y = Q(x) over the interval 4 ? x ? 4? (A)[img]13457[/img] (B)[img]13458[/img] (C)[img]13459[/img] (D)[img]13460[/img] (E)[img]13461[/img]

USA
AMC 12 2002

B

1 The arithmetic mean of the nine numbers in the set {9, 99, 999, 9999, ..., 999999999} is a 9-digit number M , all of whose digits are distinct. The number M does not contain the digit (A) 0 (B) 2 (C) 4 (D) 6 (3x when x = 4? (A) 0 (B) 1 (C) 10 (D) 11 (E) 12 3n + 2 a prime number? (D) more than two, but ?nitely many 3 For how many positive integers n is n2 (A) none (B) one (E) in?nitely many (C) two (E) 8 (3x 2)4x + 1 2 What is the value of 2)(4x + 1)

1 1 1 1 4 Let n be a positive integer such that + + + is an integer. Which of the following 2 3 7 n statements is not true? (A) 2 divides n (E) n > 84 (B) 3 divides n (C) 6 divides n (D) 7 divides n

5 Let v , w, x, y , and z be the degree measures of the ?ve angles of a pentagon. Suppose v < w < x < y < z and v , w, x, y , and z form an arithmetic sequence. Find the value of x. (A) 72 (B) 84 (C) 90 (D) 108 (E) 120 6 Suppose that a and b are are nonzero real numbers, and that the equation x2 + ax + b = 0 has positive solutions a and b. Then the pair (a, b) is (A) ( 2, 1) (B) ( 1, 2) (C) (1, 2) (D) (2, 1) (E) (4, 4) 7 The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares? (A) 50 (B) 77 (C) 110 (D) 149 (E) 194 8 Suppose July of year N has ?ve Mondays. Which of the following must occur ?ve times in August of year N ? (Note: Both months have 31 days.) (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday

USA
AMC 12 2002

9 If a, b, c, and d are positive real numbers such that a, b, c, d form an increasing arithmetic a sequence and a, b, d form a geometric sequence, then is d 1 1 1 1 1 (A) (B) (C) (D) (E) 12 6 4 3 2 10 How many di?erent integers can be expressed as the sum of three distinct members of the set {1, 4, 7, 10, 13, 16, 19}? (A) 13 (B) 16 (C) 24 (D) 30 (E) 35 11 The positive integers A, B , A primes is (A) even (E) prime (B) divisible by 3 n 20 B , and A + B are all prime numbers. The sum of these four (C) divisible by 5 (D) divisible by 7

12 For how many integers n is (A) 1 (B) 2 (C) 3

the square of an integer? n (D) 4 (E) 10

13 The sum of 18 consecutive positive integers is a perfect square. The smallest possible value of this sum is (A) 169 (B) 225 (C) 289 (D) 361 (E) 441 14 Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? (A) 8 (B) 9 (C) 10 (D) 12 (E) 16 15 How many four-digit numbers N have the property that the three-digit number obtained by removing the leftmost digit is one ninth of N ? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 16 Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? 1 1 1 7 2 (A) (B) (C) (D) (E) 12 3 2 12 3 17 Andys lawn has twice as much area as Beths lawn and three times as much area as Carlos lawn. Carlos lawn mower cuts half as fast as Beths mower and one third as fast as Andys mower. If they all start to mow their lawns at the same time, who will nish rst? (A) Andy (B) Beth (E) All three tie. (C) Carlos (D) Andy and Carlos tie for ?rst.

USA
AMC 12 2002

18 A point P is randomly selected from the rectangular region with vertices (0, 0), (2, 0), (2, 1), (0, 1). What is the probability that P is closer to the origin than it is to the point (3, 1)? 1 2 3 4 (A) (B) (C) (D) (E) 1 2 3 4 5 19 If a, b, and c are positive real numbers such that a(b + c) = 152, b(c + a) = 162, and c(a + b) = 170, then abc is (A) 672 (B) 688 (C) 704 (D) 720 (E) 750 20 Let 4XOY be a right-angled triangle with m\XOY = 90 . Let M and N be the midpoints of legs OX and OY , respectively. Given that XN = 19 and Y M = 22, ?nd XY . (A) 24 (B) 26 (C) 28 (D) 30 2002, let if n is divisible by 13 and 14 if n is divisible by 11 and 14 if n is divisible by 11 and 13 otherwise (E) 32 21 For all positive integers n less than 8 > 11 > > > <13 an = > 14 > > > :0 Calculate (A) 448
2001 X n=1

an .

(B) 486

(C) 1560

(D) 2001

(E) 2002

22 For all integers n greater than 1, de?ne an =

1 . Let b = a2 + a3 + a4 + a5 and logn 2002 c = a10 + a11 + a12 + a13 + a14 . Then b c equals 1 1 1 (A) 2 (B) 1 (C) (D) (E) 2002 1001 2

23 In 4ABC , we have AB = 1 and AC = 2. Side BC and the median from A to BC have the same length. What is BC ? p p p p 1+ 2 1+ 3 3 (A) (B) (C) 2 (D) (E) 3 2 2 2 24 A convex quadrilateral ABCD with area 2002 contains a point P in its interior such that P A = 24, P B = 32, P C = 28, and P D = 45. FInd the perimeter of ABCD. ? ? ? ? p p p p p (A) 4 2002 (B) 2 8465 (C) 2 48 + 2002 (D) 2 8633 (E) 4 36 + 113

USA
AMC 12 2002

25 Let f (x) = x2 + 6x + 1, and let R denote the set of points (x, y ) in the coordinate plane such that f (x) + f (y ) ? 0 and f (x) f (y ) ? 0 The area of R is closest to (A) 21 (B) 22 (C) 23 (D) 24 (E) 25

USA
AMC 12 2003

A
1 What is the di?erence between the sum of the ?rst 2003 even counting numbers and the sum of the ?rst 2003 odd counting numbers? (A) 0 (B) 1 (C) 2 (D) 2003 (E) 4006 2 Members of the Rockham Soccer League buy socks and T-shirts. Socks cost U N IQ3d3e2262c QIN U 4 per pair and each T-shirt costs U N IQ3d3e2262c QIN U 5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is U N IQ3d3e2262c QIN U 2366, how many members are in the League? (A) 77 (B) 91 (C) 143 (D) 182 (E) 286 3 A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed? (A) 4.5 (B) 9 (C) 12 (D) 18 (E) 24 4 It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? (A) 3 (A) 10 (B) 3.125 (B) 11 (C) 3.5 (C) 12 (D) 4 (D) 13 (E) 4.5 (E) 14 5 The sum of the two 5-digit numbers AM C 10 and AM C 12 is 123422. What is A + M + C ? 6 De?ne x~y to be |x not true? y | for all real numbers x and y . Which of the following statements is

(A) x~y = y ~x for all x and y (B) 2(x~y ) = (2x)~(2y ) for all x and y (C) x~0 = x for all x (D) x~x = 0 for all x (E) x~y > 0 if x 6= y 7 How many non-congruent triangles with perimeter 7 have integer side lengths? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 8 What is the probability that a randomly drawn positive factor of 60 is less than 7? 1 1 1 1 1 (A) (B) (C) (D) (E) 10 6 4 3 2

USA
AMC 12 2003

9 A set S of points in the xy -plane is symmetric about the origin, both coordinate axes, and the line y = x. If (2, 3) is in S , what is the smallest number of points in S ? (A) 1 (B) 2 (C) 4 (D) 8 (E) 16 10 Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of 3 : 2 : 1, respectively. Due to some confusion they come at di?erent times to claim their prizes, and each assumes he is the rst to arrive. If each takes what he believes to be his correct share of candy, what fraction of the candy goes unclaimed? 1 1 2 5 5 (A) (B) (C) (D) (E) 18 6 9 18 12 11 A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the square and B be the area of the circle circumscribed about the triangle. Find A/B . p 9 3 27 3 6 (A) (B) (C) (D) (E) 1 16 4 32 8 12 Sally has ve red cards numbered 1 through 5 and four blue cards numbered 3 through 6. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12 13 The polygon enclosed by the solid lines in the gure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen ?nedashed=linetype(quot;4 4quot;);

?lldraw((1,1)–(2,1)–(2,2)–(4,2)–(4,3)–(1,3)–cycle,grey,black+linewidth(.8pt)); draw((0,1)–(0,3)– (1,3)–(1,4)–(4,4)–(4,3)– (5,3)–(5,2)–(4,2)–(4,1)–(2,1)–(2,0)–(1,0)–(1,1)–cycle,?nedashed); draw((0,2)– (2,2)–(2,4),?nedashed); draw((3,1)–(3,4),?nedashed); label(quot;36;136;quot;,(1.5,0.5)); draw(circle((1.5,0.5), label(quot;36;236;quot;,(2.5,1.5)); draw(circle((2.5,1.5),.17)); label(quot;36;336;quot;,(3.5,1.5)); draw(circle((3.5,1.5),.17)); label(quot;36;436;quot;,(4.5,2.5)); draw(circle((4.5,2.5),.17)); label(quot;36;536;quot;,(3.5,3.5)); draw(circle((3.5,3.5),.17)); label(quot;36;636;quot;,(2.5,3.5)); draw(circle((2.5,3.5),.17)); label(quot;36;736;quot;,(1.5,3.5)); draw(circle((1.5,3.5),.17)); label(quot;36;836;quot;,(0.5,2.5)); draw(circle((0.5,2.5),.17)); label(quot;36;936;quot;,(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

USA
AMC 12 2003

14 Points K , L, M , and N lie in the plane of the square ABCD so that AKB , BLC , CM D, and DN A are equilateral triangles. If ABCD has an area of 16, nd the area of KLM N .

[asy]unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A–N–K–A–B–K–L–B–C–L–M–C–D–M–N–D–A); label(quot;36;A36;quot;,A,SE); label(quot;36;B36;quo label(quot;36;C36;quot;,C,NW); label(quot;36;D36;quot;,D,NE); label(quot;36;K36;quot;,K,NNW); label(quot;36;L36;quot;,L,E); label(quot;36;M36;quot;,M,S); label(quot;36;N36;quot;,N,W);[/asy] p p (A) 32 (B) 16 + 16 3 (C) 48 (D) 32 + 16 3 (E) 64 15 A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)); ?lldraw(Circle((0,.866),.5),grey,black); label(quot;1quot;,(0,.866),S); ?lldraw(Circle((0,0),1),white,black); draw((-.5,.866)–(.5,.866),dashed); clip((-1,0)–(1,0)–(1,2)–(-1,2)–cycle); draw((-1,0)–(1,0)); label(quot;2quot;,(0,0),S);[/asy] p p p p p 1 3 3 3 3 3 1 1 1 1 (A) ? (B) ? (C) ? (D) + ? (E) + ? 6 4 4 12 4 24 4 24 4 12 16 A point P is chosen at random in the interior of equilateral triangle ABC . What is the probability that 4ABP has a greater area than each of 4ACP and 4BCP ? 1 1 1 1 2 (A) (B) (C) (D) (E) 6 4 3 2 3 17 Square ABCD has sides of length 4, and M is the midpoint of CD. A circle with radius 2 and center M intersects a circle with raidus 4 and center A at points P and D. What is the distance from P to AD? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(Circle((2,0),2)); draw(Circle((0,4),4)); clip(scale(4)*unitsquare); draw(scale(4)*unitsquare); ?lldraw(Circle((2,0),0.07)); ?lldraw(Circle((3.2,1.6),0.07)); label(quot;36;A36;quot;,(0,4),NW); label(quot;36;B36;quot;,(4,4),NE); label(quot;36;C36;quot;,(4,0),SE); label(quot;36;D36;quot;,(0,0),SW); label(quot;36;M36;quot;,(2,0),S); label(quot;36;P36;quot;,(3.2,1.6),N);[/asy] p 16 13 7 (A) 3 (B) (C) (D) 2 3 (E) 5 4 2 18 Let n be a 5-digit number, and let q and r be the quotient and remainder, respectively, when n is divided by 100. For how many values of n is q + r divisible by 11? (A) 8180 (B) 8181 (C) 8182 (D) 9000 (E) 9090

USA
AMC 12 2003

19 A parabola with equation y = ax + bx + c is reected about the x-axis. The parabola and its reection are translated horizontally ve units in opposite directions to become the graphs of y = f (x) and y = g (x), respectively. Which of the following describes the graph of y = (f + g )(x)? (A) a parabola tangent to the x - axis (B) a parabola not tangent to the x - axis (C) a horizontal line (D) a non - horizontal line (E) the graph of a cubic function 20 How many 15-letter arrangements of 5 A’s, 5 B’s, and 5 C’s have no A’s in the ?rst 5 letters, no B’s in the next 5 letters, and no C’s in the last 5 letters? 5 ? ◆3 X 5 15! (A) (E) 315 (B) 35 · 25 (C) 215 (D) (5!)3 k
k=0

21 The graph of the polynomial P (x) = x5 + ax4 + bx3 + cx2 + dx + e has ?ve distinct x-intercepts, one of which is at (0, 0). Which of the following coe cients cannot be zero? (A) a (B) b (C) c (D) d (E) e 22 Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts at (0, 0) and each of its steps is either right or up, both equally likely. Object B starts at (5, 7) and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet? (A) 0.10 (A) 504 (B) 0.15 (B) 682 (C) 0.20 (C) 864 (D) 0.25 (D) 936 (E) 0.30 (E) 1008 23 How many perfect squares are divisors of the product 1! · 2! · 3! · · · 9!? 24 If a (A) b > 1, what is the largest possible value of loga (a/b) + logb (b/a)? 2

(B) 0 (C) 2 (D) 3 (E) 4 p 25 Let f (x) = ax2 + bx. For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the same set? (A) 0 (B) 1 (C) 2 (D) 3 (E) in?nitely many

USA
AMC 12 2003

B
1 Which of the following is the same as 2 4 + 6 8 + 10 12 + 14 ? 3 6 + 9 12 + 15 18 + 21 (A) 1 (B) 2 3 (C) 2 3 (D) 1 (E) 14 3

2 Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs U N IQcb3a90f f 6 QIN U 1 more than a pink pill, and Als pills cost a total of U N IQcb3a90f f 6 QIN U 546 for the two weeks. How much does one green pill cost? (A) U N IQcb3a90f f 6 QIN U 7 (B) U N IQcb3a90f f 6 QIN U 14 (C) U N IQcb3a90f f 6 QIN U 19 (D) U N IQcb3a90f f 6 QIN U 20 (E) U N IQcb3a90f f 6 QIN U 39 3 Rose ?lls each of the rectangular regions of her rectangular ?ower bed with a di?erent type of ?ower. The lengths, in feet, of the rectangular regions in her ?ower bed are as shown in the gure. She plants one ?ower per square foot in each region. Asters cost U N IQc1d4ba9a8 QIN U 1 each, begonias U N IQc1d4ba9a8 QIN U 1.50 each, cannas U N IQc1d4ba9a8 QIN U 2 each, dahlias U N IQc1d4ba9a8 QIN U 2.50 each, and Easter lilies U N IQc1d4ba9a8 QIN U 3 each. What is the least possible cost, in dollars, for her garden? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((6,0)–(0,0)–(0,1)–(6,1)); draw((0,1)–(0,6)–(4,6)–(4,1)); draw((4,6)–(11,6)–(11,3)–(4,3)); draw((11,3)–(11,0)–(6,0)–(6,3));

label(quot;1quot;,(0,0.5),W); label(quot;5quot;,(0,3.5),W); label(quot;3quot;,(11,1.5),E); label(quot;3quot;,(11,4.5),E); label(quot;4quot;,(2,6),N); label(quot;7quot;,(7.5,6),N); label(quot;6quot;,(3,0),S label(quot;5quot;,(8.5,0),S);[/asy](A) 108 (B) 115 (C) 132 (D) 144 (E) 156 4 Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? (A) 0.75 (B) 0.8 (C) 1.35 (D) 1.5 (E) 3 5 Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is 4 : 3. The horizontal length of a 27-inch television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt));

USA
AMC 12 2003

draw((0,0)–(4,0)–(4,3)–(0,3)–(0,0)–(4,3)); ?ll((0,0)–(4,0)–(4,3)–cycle,mediumgray); label(rotate(aTan(3.0/4.0 label(rotate(90)*quot;Heightquot;,(4,1.5),E); label(quot;Lengthquot;,(2,0),S);[/asy](A) 20 (B) 20.5 ( 6 The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible ?rst term? p p p p 2 3 3 (A) 3 (B) (C) (D) 3 (E) 3 3 3 7 Penniless Petes piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is U N IQb34ae9b3c QIN U 8.35. It does not necessarily contain coins of all three types. What is the di?erence between the largest and smallest number of dimes that could be in the bank? (A) 0 (B) 13 (C) 37 (D) 64 (E) 83 8 Let |(x) denote the sum of the digits of the positive integer x. For example, |(8) = 8 and |(123) = 1 + 2 + 3 = 6. For how many two-digit values of x is |(|(x)) = 3? (A) 3 (B) 4 (C) 6 (D) 9 (E) 10 9 Let f be a linear function for which f (6) (A) 12 (B) 18 (C) 24 (D) 30 f (2) = 12. What is f (12) (E) 36 f (2)?

10 Several gures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the ve positions shown. How many non-congruent gures can be constructed in this way? [asy]unitsize(2cm); pair A=dir(306); pair B=dir(234); pair C=dir(162); pair D=dir(90); pair E=dir(18); draw(A–B–C–D–E–cycle,linewidth(.8pt)); draw(E–rotate(60,D)*E–D–rotate(60,C)*D–C–rotate(60,B)*C– B–rotate(60,A)*B–A–rotate(60,E)*A–cycle,linetype(quot;4 4quot;)); label(quot;36;A36;quot;,A,SE); label(quot;36;B36;quot;,B,SW); label(quot;36;C36;quot;,C,WNW); label(quot;36;D36;quot;,D,N); label(quot;36;E36;quot;,E,ENE);[/asy](A) 1 (B) 2 (C) 3 (D) 4 11 Cassandra sets her watch to the correct time at noon. At the actual time of 1: 00 PM, she notices that her watch reads 12: 57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch rst reads 10: 00 PM? (A) 10: 22 PM and 24 seconds (B) 10: 24 PM (C) 10: 25 PM (D) 10: 27 PM (E) 10: 30 PM 12 What is the largest integer that is a divisor of (n + 1)(n + 3)(n + 5)(n + 7)(n + 9) for all positive even integers n? (A) 3 (B) 5 (C) 11 (D) 15 (E) 165

USA
AMC 12 2003

13 An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly ll the cone. Assume that the melted ice cream occupies 75% of the volume of the frozen ice cream. What is the ratio of the cones height to its radius? (A) 2 : 1 (B) 3 : 1 (C) 4 : 1 (D) 16 : 3 (E) 6 : 1 14 In rectangle ABCD, AB = 5 and BC = 3. Points F and G are on CD so that DF = 1 and GC = 2. Lines AF and BG intersect at E . Find the area of 4AEB . [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A–B–C–D–A–E–B); label(quot;36;A36;quot;,A,SW); label(quot;36;B36;quot;,B,SE); label(quot;36;C36;quot;,C,NE); label(quot;36;D36;quot;,D,NW); label(quot;36;E36;quot;,E,N); label(quot;36;F36;quot;,F,SE); label(quot;36;G36;quot;,G,SW); label(quot;36;B36;quot;,B,SE); label(quot;1quot;,midpoint(D–F),N); label(quot;2quot;,midpoint(G–C),N); label(quot;3quot;,midpoint(B– 21 C),E); label(quot;3quot;,midpoint(A–D),W); label(quot;5quot;,midpoint(A–B),S);[/asy](A) 10 (B) 2 15 A regular octagon ABCDEF GH has an area of one square unit. What is the area of the rectangle ABEF ? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(6pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);

draw(A–B–C–D–E–F–G–H–cycle); label(quot;36;A36;quot;,A,NNW); label(quot;36;B36;quot;,B,NNE); label(quot;36;C36;quot;,C,ENE); label(quot;36;D36;quot;,D,ESE); label(quot;36;E36;quot;,E,SSE); label(quot;36;F36;quot;,F,SSW); label(quot;36;G36;quot;,G,WSW); label(quot;36;H36;quot;,H,WNW);[/asy] p p p p 2 2 1 1+ 2 (B) (C) 2 1 (D) (E) 2 4 2 4 16 Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype(quot;4 4quot;); dotfactor=3; pair A=(-2,0), B=(2,0);

?ll(Arc((0,0),2,0,180)–cycle,mediumgray); ?ll(Arc((-1,0),1,0,180)–cycle,white); ?ll(Arc((0,0),1,0,180)– cycle,white); ?ll(Arc((1,0),1,0,180)–cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,1 cycle);

USA
AMC 12 2003

dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)–(-2,-0.3),gray); draw((-1,-0.1)–(-1,-0.3),gray); draw((1,-0.1)–(1,-0.3),gray); draw((2,0.1)–(2,-0.3),gray); label(quot;36;A36;quot;,A,W); label(quot;36;B36;quot;,B,E); label(quot;1quot;,(-1.5,-0.1),S); p p label(quot;2quot;,(0,-0.1),S); label(quot;1quot;,(1.5,-0.1),S);[/asy](A) ? 3 (B) ? 2 p 7 3 (E) ? 6 2 17 If log(xy 3 ) = 1 and log(x2 y ) = 1, what is log(xy )? 1 1 3 (A) (B) 0 (C) (D) (E) 1 2 2 5 18 Let x and y be positive integers such that 7x5 = 11y 13 . The minimum possible value of x has a prime factorization ac bd . What is a + b + c + d? (A) 30 (B) 31 (C) 32 (D) 33 (E) 34 19 Let S be the set of permutations of the sequence 1, 2, 3, 4, 5 for which the rst term is not 1. A permutation is chosen randomly from S . The probability that the second term is 2, in lowest terms, is a/b. What is a + b? (A) 5 (B) 6 (C) 11 (D) 16 (E) 19 20 Part of the graph of f (x) = x3 + bx2 + cx + d is shown. What is b? [asy]import graph; unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; real y(real x) return (x-1)*(x+1)*(x-2); path bounds=(-1.5,-1)–(1.5,-1)–(1.5,2.5)–(-1.5,2.5)–cycle; p ?+ 2 (C) 2

pair[] points=(-1,0),(0,2),(1,0); draw(bounds,white); draw(graph(y,-1.5,1.5)); drawline((0,0),(1,0)); drawline((0,0),(0,1)); dot(points); label(quot;36;(-1,0)36;quot;,(-1,0),SE); label(quot;36;(1,0)36;quot;,(1,0),SW label(quot;36;(0,2)36;quot;,(0,2),NE); clip(bounds);[/asy](A) 4 (B) 2 (C) 0 (D) 2 (E) 4 21 An object moves 8 cm in a straight line from A to B , turns at an angle ?, measured in radians and chosen at random from the interval (0, ? ), and moves 5 cm in a straight line to C . What is the probability that AC < 7? 1 1 1 1 1 (A) (B) (C) (D) (E) 6 5 4 3 2 22 Let ABCD be a rhombus with AC = 16 and BD = 30. Let N be a point on AB , and let P and Q be the feet of the perpendiculars from N to AC and BD, respectively. Which of

USA
AMC 12 2003

the following is closest to the minimum possible value of P Q? [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C; pair Np=waypoint(B–A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D); draw(A–B–C–D–cycle); draw(A–C); draw(B–D); draw(Np–Q); draw(Np–P); label(quot;36;D36;quot;,D,SW); label(quot;36;C36;quot;,C,SE); label(quot;36;B36;quot;,B,NE); label(quot;36;A36;quot;,A,NW); label(quot;36;N36;quot;,Np,N); label(quot;36;P36;quot;,P,SW); label(quot;36;Q36;quot;,Q,SSE); draw(rightanglemark(Np,P,C,2)); draw(rightanglemark(Np,Q,D,2));[/asy](A) 6.5 (B) 6.75 (C) 7 23 The number of x-intercepts on the graph of y = sin(1/x) in the interval (0.0001, 0.001) is closest to (A) 2900 (B) 3000 (C) 3100 (D) 3200 (E) 3300 24 Positive integers a, b, and c are chosen so that a < b < c, and the system of equations 2x + y = 2003 and y = |x (A) 668 (B) 669 (C) 1002 (D) 2003 a| + |x b| + |x c|

has exactly one solution. What is the minimum value of c? (E) 2004 25 Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? 1 1 1 1 1 (A) (B) (C) (D) (E) 36 24 18 12 9

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AMC 12 2004

A

1 Alicia earns U N IQ1b36e5481 QIN U 20 per hour, of which 1.45% is deducted to pay local taxes. How many cents per hour of Alicia’s wages are used to pay local taxes? (A) 0.0029 (B) 0.029 (C) 0.29 (D) 2.9 (E) 29 2 On the AMC 12, each correct answer is worth 6 points, each incorrect answer is worth 0 points, and each problem left unanswered is worth 2.5 points. If Charlyn leaves 8 of the 25 problems unanswered, how many of the remaining problems must she answer correctly in order to score at least 100? (A) 11 (A) 33 (B) 13 (B) 49 (C) 14 (C) 50 (D) 16 (D) 99 (E) 17 (E) 100 3 For how many ordered pairs of positive integers (x, y ) is x + 2y = 100? 4 Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha’s daughters and grand-daughters have no children? (A) 22 (B) 23 (C) 24 (D) 25 (E) 26 5 The graph of the line y = mx + b is shown. Which of the following is true? [asy]import math; unitsize(8mm); defaultpen(linewidth(1pt)+fontsize(6pt)); dashed=linetype(quot;4 4quot;)+linewidth(.8pt); draw((-2,-2.5)–(-2,2.5)–(2.5,2.5)–(2.5,-2.5)–cycle,white); label(quot;36;-136;quot;,(-1,0),SW); label(quot;36;136;quot;,(1,0),SW); label(quot;36;236;quot;,(2,0),SW); label(quot;36;136;quot;,(0,1),NE); label(quot;36;236;quot;,(0,2),NE); label(quot;36;-136;quot;,(0,1),SW); label(quot;36;-236;quot;,(0,-2),SW); drawline((0,0),(1,0)); drawline((0,0),(0,1)); drawline((0,0.8),(1.8,0)); drawline((1,0),(1,1),dashed); drawline((2,0),(2,1),dashed); drawline((-1,0),(-1,1),dashed); drawline((0,1),(1,1),dashed); drawline((0,2),(1,2),dashed); drawline((0,-1),(1,-1),dashed); drawline((0,-2),(1,-2),dashed);[/asy] (A) mb < 1 (B) 1 < mb < 0 (C) mb = 0 (D) 0 < mb < 1 (E) mb > 1 6 Let U = 2 ? 20042005 , V = 20042005 , W = 2003 ? 20042004 , X = 2 ? 20042004 , Y = 20042004 and Z = 20042003 . Which of the following is the largest? (A) U V (B) V W (C) W X (D) X Y (E) Y Z 7 A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a

USA
AMC 12 2004

discard pile. The game ends when some player runs out of tokens. Players A, B , and C start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game? (A) 36 (B) 37 (C) 38 (D) 39 (E) 40 8 In the overlapping triangles 4ABC and 4ABE sharing common side AB , \EAB and \ABC are right angles, AB = 4, BC = 6, AE = 8, and AC and BE intersect at D. What is the di?erence between the areas of 4ADE and 4BDC ? (A) 2 (B) 4 (C) 5 (D) 8 (E) 9 9 A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by 25% without altering the volume, by what percent must the height be decreased? (A) 10% (A) 7 (B) 25% (B) 72 (C) 36% (D) 74 (D) 50% (E) 75 (E) 60% 10 The sum of 49 consecutive integers is 75 . What is their median? (C) 73 11 The average value of all the pennies, nickels, dimes, and quarters in Paula’s purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 12 Let A = (0, 9) and B = (0, 12). Points A0 and B 0 are on the line y = x, and AA0 and BB 0 intersect at C = (2, 8). What is the length of A0 B 0 ? p p p (A) 2 (B) 2 2 (C) 3 (D) 2 + 2 (E) 3 2 13 Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be or 1. How many distinct lines pass through at least two members of S ? (A) 8 (B) 20 (C) 24 (D) 27 (E) 36 1, 0,

14 A sequence of three real numbers forms an arithmetic progression with a ?rst term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression? (A) 1 (B) 4 (C) 36 (D) 49 (E) 81 15 Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They ?rst meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their ?rst meeting point. Each girl runs at a constant speed. What is the length of the track in meters? (A) 250 (B) 300 (C) 350 (D) 400 (E) 500

USA
AMC 12 2004

16 The set of all real numbers x for which log2004 (log2003 (log2002 (log2001 x))) is de?ned as {x|x > c}. What is the value of c? (A) 0 (B) 20012002 (C) 20022003

(D) 20032004

(E) 20012002

2003

17 Let f be a function with the following properties: (i) f (1) = 1, and (ii) f (2n) = n ? f (n), for any positive integer n. What is the value of f (2100 )? (A) 1 (B) 299 (C) 2100 (D) 24950 (E) 29999 18 Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicricle from C intersects side AD at E . What is the length of CE ? p p p p 5 2+ 5 (B) 5 (C) 6 (D) (E) 5 5 (A) 2 2 19 Circles A, B and C are externally tangent to each other, and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B ? p p 2 3 7 8 1+ 3 (A) (B) (C) (D) (E) 3 2 8 9 3 20 Select numbers a and b between 0 and 1 independently and at random, and let c be their sum. Let A, B and C be the results when a, b and c, respectively, are rounded to the nearest integer. What is the probability that A + B = C ? 1 1 1 2 3 (A) (B) (C) (D) (E) 4 3 2 3 4 21 If
1 X

cos2n ? = 5, what is the value of cos 2?? 2 (B) 5 (C) p 5 5 (D) 3 5 (E) 4 5

n=0

1 (A) 5

22 Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere? p p p p 30 69 123 52 (A) 3 + (B) 3 + (C) 3 + (D) (E) 3 + 2 2 2 3 4 9

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AMC 12 2004

23 A polynomial P (x) = c2004 x2004 + c2003 x2003 + ... + c1 x + c0 has real coe cients with c2004 6= 0 and 2004 distinct complex zeroes zk = ak + bk i, 1 ? k ? 2004 with ak and bk real, a1 = b1 = 0, and
2004 X k=1

ak =

2004 X k=1

bk .

Which of the following quantities can be a nonzero number? (A) c0 (B) c2003 (C) b2 b3 ...b2004 (D)
2004 X k=1

ak

(E)

2004 X k=1

ck

24 A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover AB . What is the area of S ? p p p 8? 3 10? p (B) (C) 3? (D) 3 (E) 4? 2 3 (A) 2? + 3 3 2 3 25 For each integer n 4, let an denote the base-n number 0.133n . The product a4 a5 ...a99 can m be expressed as , where m and n are positive integers and n is as small as possible. What n! is the value of m? (A) 98 (B) 101 (C) 132 (D) 798 (E) 962

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AMC 12 2004

B

1 At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her ?fth practice she made 48 free throws. How many free throws did she make at the ?rst practice? (A) 3 (B) 6 (C) 9 (D) 12 (E) 15 2 In the expression c · ab d, the values of a, b, c, and d are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result? (A) 5 (A) 8 (B) 6 (B) 9 (C) 8 (C) 10 (D) 9 (D) 11 (E) 10 (E) 12 3 If x and y are positive integers for which 2x 3y = 1296, what is the value of x + y ? 4 An integer x, with 10 ? x ? 99, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of x is a 7? 1 1 19 2 1 (A) (B) (C) (D) (E) 9 5 90 9 3 5 On a trip from the United States to Canada, Isabella took d U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had d Canadian dollars left. What is the sum of the digits of d? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 6 Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17 7 A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? (A) 200 + 25? (B) 100 + 75? (C) 75 + 100? (D) 100 + 100? (E) 100 + 125? 8 A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain? (A) 5 (B) 8 (C) 9 (D) 10 (E) 11

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AMC 12 2004

9 The point ( 3, 2) is rotated 90 clockwise around the origin to point B . Point B is then re?ected over the line y = x to point C . What are the coordinates of C ? (A) ( 3, 2) (B) ( 2, 3) (C) (2, 3) (D) (2, 3) (E) (3, 2) 10 An annulus is the region between two concentric circles. The concentric circles in the gure have radii b and c, with b > c. Let OX be a radius of the larger circle, let XZ be tangent to the smaller circle at Z , and let OY be the radius of the larger circle that contains Z . Let a = XZ , d = Y Z , and e = XY . What is the area of the annulus? (A) ? a2 (B) ? b2 (C) ? c2 pen(linewidth(.8pt)); dotfactor=3; (D) ? d2 (E) ? e2 [asy]unitsize(1.4cm); default-

real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z– (Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points=X,O,Y,Z; ?lldraw(Circle(O,r2),mediumgray,black); ?lldraw(Circle(O,r1),white,black); dot(points); draw(X–Y–O–cycle–Z); label(quot;36;O36;quot;,O,SSW,fontsize(10pt)); label(quot;36;Z36;quot;,Z,SW,fontsize(10pt)); label(quot;36;Y36;quot;,Y,N,fontsize(10pt)); label(quot;36;X36;quot;,X,NE,fontsize(10pt)); defaultpen(fontsize(8pt));

label(quot;36;c36;quot;,midpoint(O–Z),W); label(quot;36;d36;quot;,midpoint(Z–Y),W); label(quot;36;e36;quo Y),NE); label(quot;36;a36;quot;,midpoint(X–Z),N); label(quot;36;b36;quot;,midpoint(O–X),SE);[/asy] 11 All the students in an algebra class took a 100-point test. Five students scored 100, each student scored at least 60, and the mean score was 76. What is the smallest possible number of students in the class? (A) 10 (B) 11 (C) 12 (D) 13 (E) 14 12 In the sequence 2001, 2002, 2003, . . ., each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is 2001 + 2002 2003 = 2000. What is the 2004th term in this sequence? (A) (A) 2004 2 (B) (B) 1 2
1

(C) 0 (C) 0

(D) 4003 (D) 1 (E) 2

(E) 6007

13 If f (x) = ax + b and f

(x) = bx + a with a and b real, what is the value of a + b?

14 In 4ABC , AB = 13, AC = 5, and BC = 12. Points M and N lie on AC and BC , respectively, with CM = CN = 4. Points J and K are on AB so that M J and N K are perpendicular to AB . What is the area of pentagon CM JKN ? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair C=(0,0), B=(12,0), A=(0,5), M=(0,4), Np=(4,0); pair K=foot(Np,A,B), J=foot(M,A,B);

USA
AMC 12 2004

draw(A–B–C–cycle); draw(M–J); draw(Np–K); label(quot;36;C36;quot;,C,SW); label(quot;36;A36;quot;,A,NW); label(quot;36;B36;quot;,B,SE); label(quot;36;N36;quot;,Np,S); label(quot;36;M36;quot;,M,W); label(quot;36;J36;quot;,J,NE); 81 205 240 label(quot;36;K36;quot;,K,NE);[/asy](A) 15 (B) (C) (D) (E) 20 5 12 13 15 The two digits in Jacks age are the same as the digits in Bills age, but in reverse order. In ?ve years Jack will be twice as old as Bill will be then. What is the di?erence in their current ages? (E) 45 p 16 A function f is de?ned by f (z ) = iz ?, where i = 1 and z ? is the complex conjugate of z . How many values of z satisfy both |z | = 5 and f (z ) = z ? (A) 0 (B) 1 (C) 2 (D) 4 (E) 8 17 For some real numbers a and b, the equation 8x3 + 4ax2 + 2bx + a = 0 has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 5, what is the value of a? (A) 256 (B) 64 (C) 8 (D) 64 (E) 256 18 Points A and B are on the parabola y = 4x2 + 7x 1, and the origin is the midpoint of AB . What is the length of AB ? p p p p 2 (A) 2 5 (B) 5 + (C) 5 + 2 (D) 7 (E) 5 2 2 19 A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? p p (A) 6 (B) 4 5 (C) 9 (D) 10 (E) 6 3 20 Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? 1 5 3 7 1 (A) (B) (C) (D) (E) 4 16 8 16 2 21 The graph of 2x2 + xy + 3y 2 20y + 40 = 0 is an ellipse in the rst quadrant of the y xy -plane. Let a and b be the maximum and minimum values of over all points (x, y ) on x the ellipse. What is the value of a + b? 11x (A) 9 (B) 18 (C) 27 (D) 36

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AMC 12 2004

(A) 3

(B)

p

10

(C)

7 2

(D)

9 2 50 d g

p (E) 2 14

22 The square b e h c f 2

is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of g ? (A) 10 (B) 25 (C) 35 (D) 62 (E) 136 23 The polynomial x3 2004x2 + mx + n has integer coe cients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of n are possible? (A) 250,000 (B) 250,250 (C) 250,500 (D) 250,750 (E) 251,000 24 In 4ABC , AB = BC , and BD is an altitude. Point E is on the extension of AC such that BE = 10. The values of tan CBE , tan DBE , and tan ABE form a geometric progression, and the values of cot DBE , cot CBE , cot DBC form an arithmetic progression. What is the area of 4ABC ? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(3,0), A=(-3,0), B=(0, 8), Ep=(6,0); draw(A–B–Ep–cycle); draw(D–B–C); label(quot;36;A36;quot;,A,S); label(quot;36;D36;quot;,D,S); label(quot;36;C36;quot;,C,S); lap 50 bel(quot;36;E36;quot;,Ep,S); label(quot;36;B36;quot;,B,N);[/asy](A) 16 (B) (C) 10 3 3 25 Given that 22004 is a 604-digit number whose rst digit is 1, how many elements of the set S = {20 , 21 , 22 , . . . , 22003 } have a rst digit of 4? (A) 194 (B) 195 (C) 196 (D) 197 (E) 198 (D) 8

p

USA
AMC 12 2005

A
1 Two is 10% of x and 20% of y . What is x A. 1 B. 2 C. 5 D. 10 E. 20 2 The equations 2x + 7 = 3 and bx A. -8 B. -4 C. -2 D. 4 E. 8 3 A rectangle with a diagonal length of x is twice as long as it is wide. What is the area of the rectangle? 1 2 1 3 A. x2 B. x2 C. x2 D. x2 E. x2 4 5 2 2 4 A store normally sells windows at U N IQ380f c5ecf QIN U 100 each. This week the store is o?ering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together than rather separately? A. 100 B. 200 C. 300 D. 400 E. 500 5 The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers? A. 23 B. 24 C. 25 D. 26 E. 27 6 Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike’s house. A little later Mike started to ride his bicycle toward Josh’s house. When they met, Josh had ridden for twice the length of time as Mike and at four-?fths of Mike’s rate. How many miles had Mike ridden when they met? A. 4 B. 5 C. 6 D. 7 E. 8 7 Square EF GH is inside the square ABCD so that each side of EF p GH can be extended to pass through a vertex of ABCD. Square ABCD has side length 50 and BE = 1. What is the area of the inner square EF GH ? A. 25 B. 32 C. 36 D. 40 E. 42 8 Let A, M , and C be digits with (100A + 10M + C )(A + M + C ) = 2005 What is A? A. 1 B. 2 C. 3 D. 4 E. 5 10 = 2 have the same solution. What is the value of b? y?

USA
AMC 12 2005

9 There are two values of a for which the equation 4x2 + ax + 8x + 9 = 0 has only one solution for x. What is the sum of these values of a? A. -16 B. -8 C. 0 D. 8 E. 20 10 A wooden cube n units on a side is painted red on all six faces and then cut into n3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n? A. 3 B. 4 C. 5 D. 6 E. 7 11 How many three-digit numbers satisfy the property that the middle digit is the average of the ?rst and the last digits? A. 41 B. 42 C. 43 D. 44 E. 45 12 A line passes through A(1,1) and B(100,1000). How many other points with integer coordinates are on the line and strictly between A and B? A. 0 B. 2 C. 3 D. 8 E. 9 13 The regular 5-point star ABCDE is drawn and in each vertex, there is a number. Each A,B,C,D, and E are chosen such that all 5 of them came from set 3,5,6,7,9. Each letter is a di?erent number (so one possible ways is A=3,B=5,C=6,D=7,E=9). Let AB be the sum of the numbers in A and B. If AB,BC,CD,DE, and EA form an arithmetic sequence (not necessarily this order), ?nd the value of CD. A. 9 B. 10 C. 11 D. 12 E. 13 14 On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? 5 10 1 11 6 (A) (B) (C) (D) (E) 11 21 2 21 11 15 Let AB be a diameter of a circle and C be a point on AB with 2 · AC = BC . Let D and E be points on the circle such that DC ? AB and DE is a second diameter. What is the ratio of the area of 4DCE to the area of 4ABD? 1 1 1 1 2 (A) (B) (C) (D) (E) 6 4 3 2 3 16 Three circles of radius s are drawn in the ?rst quadrant of the xy -plane. The ?rst circle is tangent to both axes, the second is tangent to the ?rst circle and the x-axis, and the third is tangent to the ?rst circle and the y -axis. A circle of radius r > s is tangent to both axes and to the second and third circles. What is r/s?

USA
AMC 12 2005

17 A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex W ? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 320[/img ] 1 1 1 1 1 A) B) C) D) E) 12 9 8 6 4 18 Call a number ”prime-looking” if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? (A) 100 (B) 102 (C) 104 (D) 106 (E) 108

19 A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? (A) 1404 20 For each x in [0, 1], de?ne f (x) = 2x, f (x) = 2 2x,
1 if 0 ? x ? 2 ; 1 if 2 < x ? 1.

(B) 1462

(C) 1604

(D) 1605

(E) 1804

Let f [2] (x) = f (f (x)), and f [n+1] (x) = f [n] (f (x)) for each integer n 1 x in [0, 1] is f [2005] (x) = ? 2 (A) 0 (B) 2005 (C) 4010 (D) 20052

2. For how many values of

(E) 22005

21 How many ordered triples of integers (a, b, c), with a 2, b 1, and c 0, satisfy both loga b = c2005 and a + b + c = 2005? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 22 A rectangular box P is inscribed in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of its 12 edges is 112. What is r? (A) 8 (B) 10 (C) 12 (D) 14 (E) 16

23 Two distinct numbers a and b are chosen randomly from the set {2, 22 , 23 , . . . , 225 }. What is the probability that loga b is an integer? (A) 2 25 (B) 31 300 (C) 13 100 (D) 7 50 (E) 1 2

USA
AMC 12 2005

24 Let P (x) = (x 1)(x 2)(x 3). For how many polynomials Q(x) does there exist a polynomial R(x) of degree 3 such that P (Q(x)) = P (x) · R(x)? (A) 19 (B) 22 (C) 24 (D) 27 (E) 32 25 Let S be the set of all points with coordinates (x, y, z ), where x, y, and z are each chosen from the set {0, 1, 2}. How many equilateral triangles have all their vertices in S ? (A) 72 (B) 76 (C) 80 (D) 84 (E) 88

USA
AMC 12 2005

B

1 A scout troop buys 1000 candy bars at a price of ?ve for U N IQ04d27cdc9 QIN U 2. They sell all the candy bars at a price of two for U N IQ04d27cdc9 QIN U 1. What was their pro?t, in dollars? A. 100 B. 200 C. 300 D. 400 E. 500 2 A positive number x has the property that x% of x is 4. What is x? A. 2 B. 4 C. 10 D. 20 E. 40 3 Brianna is using part of the money she earned on her weekend job to buy several equallypriced CDs. She used one ?fth of her money to buy one third of the CDs. What fraction of her money will she left after she buys all the CDs? 1 A. 5 1 B. 3 2 C. 5 2 D. 3 4 E. 5 4 At the beginning of the school year, Lisa’s goal was to earn an A on at least 80% of her 50 quizzes for the year. She earned an A on 22 of the ?rst 30 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A? A. 1 B. 2 C. 3 D. 4 E. 5 5 An 8-foot by 10-foot ?oor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the ?oor are shaded? A. 80 20? B. 60 10? C. 80 10? D. 60 + 10? E. 80 + 10? 6 In 4ABC , we have AC = BC = 7 and AB = 2. Suppose that D is a point on line AB such that B lies between A and D and CD = 8. What is BD? p p A. 3 B. 2 3 C. 4 D. 5 E. 4 2

USA
AMC 12 2005

7 What is the area enclosed by the graph of |3x| + |4y | = 12? A. 6 B. 12 C. 16 D. 24 E. 25 8 For how many values of a is it true that the line y = x + a passes through the vertex of the parbola y = x2 + a2 ? A. 0 B. 1 C. 2 D. 10 E. In?nitely many 9 On a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points. What is the di?erence between the mean and the median score on this exam? A. 0 B. 1 C. 2 D. 4 E. 5 10 The ?rst term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? A. 29 B. 55 C. 85 D. 133 E. 250 11 An envelope contains eight bills: 2 ones, 2 ?ves, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is U N IQ93d42f 50e QIN U 20 or more? 1 2 3 1 2 A. B. C. D. E. 4 7 7 2 3 12 The quadratic equation x2 + mx + n = 0 has roots that are twice those of x2 + px + m = 0, and none of m, n, and p is zero. What is the value of n/p? A. 1 B. 2 C. 4 D. 8 E. 16 13 Suppose that 4x1 = 5, 5x2 = 6, 6x3 = 7, ..., 127x124 = 128. What is x1 x2 · · · x124 ? 5 7 A. 2 B. C. 3 D. E. 4 2 2 14 A circle having center (0, k ), with k > 6, is tangent to the lines y = x, y = What is the radius of this circle? p p p A. 6 2 6 B. 6 C. 6 2 D. 12 E. 6 + 6 2 x and y = 6.

15 The sum of four two-digit numbers is 221. None of the eight digits is 0 and no two of them are same. Which of the following is not included among the eight digits? A. 1 B. 2 C. 3 D. 4 E. 5 16 Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, thta contains these eight spheres? p p p p A. 2 B. 3 C. 1 + 2 D. 1 + 3 E. 3

USA
AMC 12 2005

17 How many distinct four-tuples (a, b, c, d) of rational numbers are there with a log10 2 + b log10 3 + c log10 5 + d log10 7 = 2005? A. 0 B. 1 C. 17 D. 2004 E. in?nitely many 18 Let A(2, 2) and B (7, 7) be points in the plane. De?ne R as the region in the ?rst quadrant consisting of those points C such that ABC is an actue triangle. What is the closest integer to the area of the region R (A)25 (B )39 (C )51 (D)60 (E )80 19 Let x and y be two-digit integers such that y is obtained by reversing the digits of x. The integers x and y satisfy x2 y 2 = m2 for some positive integer m. What is x + y + m? A. 88 B. 112 C. 116 D. 144 E. 154 20 Let a, b, c, d, e, f, g and h be distinct elements in the set { 7, 5, 3, 2, 2, 4, 6, 13}. What is the minimum possible value of (a + b + c + d)2 + (e + f + g + h)2 A. 30 B. 32 C. 34 D. 40 E. 50 21 A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7k divides n? A. 0 B. 1 C. 2 D. 3 E. 4 22 A sequence of complex numbers z0 , z1 , z2 , .... is de?ned by the rule zn+1 = izn zn 1. Suppose that |z0 | = 1 and z2005 = 1.

where zn is the complex conjugate of zn and i2 = How many possible values are there for z0 ? A. 1 B. 2 C. 4 D. 2005 E. 22005

23 Let S be the set of ordered triples (x, y, z ) of real numbers for which log10 (x + y ) = z and log10 (x2 + y 2 ) = z + 1. There are real numbers a and b such that for all ordered triples (x, y, z ) in S we have x3 + y 3 = a · 103z + b · 102z . What is the value of a + b? 15 29 39 A. B. C. 15 D. E. 24 2 2 2

USA
AMC 12 2005

24 All three vertices of an equilateral triangle are on the parabola y = x2 , and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of m/n, where m and n are relatively prime positive integers. What is the value of m + n? A. 14 B. 15 C. 16 D. 17 E. 18 25 Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a di?erent vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? 5 A. 256 21 B. 1024 11 C. 512 23 D. 1024 3 E. 128

USA
AMC 12 2006

A

1 Sandwiches at Joe’s Fast Food cost U N IQ92c5b4061 QIN U 3 each and sodas cost U N IQ92c5b4061 QIN U 2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? (A) 31 (B) 32 (B) 0 (C) 33 (D) 34 (D) 2h (E) 35 2 De?ne x ? y = x3 (A) h y . What is h ? (h ? h)? (C) h

(E) h3 (E) 50

3 The ratio of Mary’s age to Alice’s age is 3 : 5. Alice is 30 years old. How old is Mary? (A) 15 (B) 18 (C) 20 (D) 24 4 A digital watch displays hours and minutes with AM and P M . What is the largest possible sum of the digits in the display? (A) 17 (B) 19 (C) 21 (D) 22 (E) 23 5 Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was U N IQ302a8ddf 0 QIN U 8, and there was an additional cost of U N IQ302a8ddf 0 QIN U 2 for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 6 The 8 ? 18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is y ? [asy]unitsize(3mm); defaultpen(fontsize(10pt));

draw((0,4)–(18,4)–(18,-4)–(0,-4)–cycle); draw((6,4)–(6,0)–(12,0)–(12,-4)); label(quot;36;A36;quot;,(0,4),NW) label(quot;36;B36;quot;,(18,4),NE); label(quot;36;C36;quot;,(18,-4),SE); label(quot;36;D36;quot;,(0,4),SW); label(quot;36;y36;quot;,(3,4),S); label(quot;36;y36;quot;,(15,-4),N); label(quot;36;1836;quot;,(9,4),N) label(quot;36;1836;quot;,(9,-4),S); label(quot;36;836;quot;,(0,0),W); label(quot;36;836;quot;,(18,0),E); dot((0,4)); dot((18,4)); dot((18,-4)); dot((0,-4));[/asy] (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 7 Mary is 20% older than Sally, and Sally is 40% younger than Danielle. The sum of their ages is 23.2 years. How old will Mary be on her next birthday? (A) 7 (B) 8 (C) 9 (D) 10 (E) 11

USA
AMC 12 2006

8 How many sets of two or more consecutive positive integers have a sum of 15? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 9 Oscar buys 13 pencils and 3 erasers for U N IQ094daa1f e QIN U 1.00. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? (D) 18 (E) 20 q p 10 For how many real values of x is 120 x an integer? (A) 3 (B) 6 (C) 9 (D) 10 (E) 11 11 Which of the following describes the graph of the equation (x + y )2 = x2 + y 2 ? (A) the empty set (B) one point (C) two lines (D) a circle (E) the entire plane 12 A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outisde diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 562[/img ] (A) 171 (B) 173 (C) 182 (D) 188 (E) 210 13 The vertices 3 4 5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 563[/img ] 25? 27? (A) 12? (B) (C) 13? (D) (E) 14? 2 2 14 Two farmers agree that pigs are worth U N IQa4f 2f 37f a QIN U 300 and that goats are worth U N IQa4f 2f 37f a QIN U 210. When one farmer owes the other money, he pays the debt in pigs or goats, with “change” received in the form of goats or pigs as necessary. (For example, a U N IQa4f 2f 37f a QIN U 390 debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way? (A) U N IQa4f 2f 37f a QIN U 5 (B) U N IQa4f 2f 37f a QIN U 10 (C) U N IQa4f 2f 37f a QIN U 30 (D) U N IQa4f 2f 37f a QIN U 90 (E) U N IQa4f 2f 37f a QIN U 210 15 Suppose cos x = 0 and cos(x + z ) = 1/2. What is the smallest possible positive value of z ? ? ? ? 5? 7? (A) (B) (C) (D) (E) 6 3 2 6 6 (A) 10 (B) 12 (C) 15

USA
AMC 12 2006

16 Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent intersects the circles at C and D, respectively. Lines AB and CD intersect at E , and AE = 5. What is CD? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 564[/img ] p p 55 44 (D) 255 (E) (A) 13 (B) (C) 221 3 3 17 Square ABCD has side length s, a circle centered at E has radius r, and r and s are both rational. The circle passes through D, and D lies on BE . Point q F lies on the circle, on the same side of BE p as A. Segment AF is tangent to the circle, and AF = 9 + 5 2. What is r/s? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 565[/img ] 1 5 3 5 9 (A) (B) (C) (D) (E) 2 9 5 3 5

18 The function f has the property that for each real number x in its domain, 1/x is also in its domain and ? ◆ 1 f (x) + f = x. x What is the largest set of real numbers that can be in the domain of f ? (A) {x|x 6= 0} 0} (C) {x|x > 0} (D) {x|x 6= 1 and x 6= 0 and x 6= 1} (E) { 1, 1}

(B) {x|x <

19 Circles with centers (2, 4) and (14, 9) have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form y = mx + b with m > 0. What is b? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 554[/img ] 908 909 130 911 912 (A) (B) (C) (D) (E) 199 119 17 119 119 20 A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? 1 1 2 1 5 (A) (B) (C) (D) (E) 2187 729 243 81 243 21 Let and S1 = {(x, y ) | log10 (1 + x2 + y 2 ) ? 1 + log10 (x + y )} S2 = {(x, y ) | log10 (2 + x2 + y 2 ) ? 2 + log10 (x + y )}. (C) 100 (D) 101 (E) 102

What is the ratio of the area of S2 to the area of S1 ? (A) 98 (B) 99

USA
AMC 12 2006

22 A circle of radius r is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is r? p p p p p p p p (A) 2p 2 + p 2 3 (B) 3 3 + 2 (C) 2 6 + 3 (D) 3 2 + 6 (E) 6 2 3 23 Given a ?nite sequence S = (a1 , a2 , . . . , an ) of n real numbers, let A(S ) be the sequence ? ◆ a1 + a2 a2 + a3 an 1 + an , ,..., 2 2 2 of n 1 real numbers. De?ne A1 (S ) = A(S ) and, for each integer m, 2 ? m ? n 1, de?ne 50 Am (S ) = A(Am 1 (S p )). Suppose x > 0, and let S = (1, x, x2 , . . . , x100 ). If A100 p(S ) = (1/2 ), then p p 2 1 2 what is x? (A) 1 (B) 2 1 (C) (D) 2 2 (E) 2 2 2 24 The expression (x + y + z )2006 + (x y z )2006 is simpli?ed by expanding it and combining like terms. How many terms are in the simpli?ed expression? (A) 6018 (B) 671, 676 (C) 1, 007, 514 (D) 1, 008, 016 (E) 2, 015, 028 25 How many non-empty subsets S of {1, 2, 3, . . . , 15} have the following two properties?

(1) No two consecutive integers belong to S . (2) If S contains k elements, then S contains no number less than k . (A) 277 (B) 311 (C) 376 (D) 377 (E) 405

USA
AMC 12 2006

B

1 What is ( 1)1 + ( 1)2 + · · · + ( 1)2006 ? (A) 2006 72 (B) 1 (C) 0

(D) 1 (D) 24

(E) 2006 y ). What is 3(45)? (E) 72

2 For real numbers x and y , de?ne xy = (x + y )(x (A) (B) 27 (C) 24

3 A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? (A) 10 (B) 14 (C) 17 (D) 20 (E) 24 4 Mary is about to pay for ?ve items at the grocery store. The prices of the items are U N IQ66152ac90 QIN U 7.99, U N IQ66152ac90 QIN U 4.99, U N IQ66152ac90 QIN U 2.99, U N IQ66152ac90 QIN U 1.99, and U N IQ66152ac90 QIN U 0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the U N IQ66152ac90 QIN U 20.00 that she will receive in change? (A) 5 (B) 10 (C) 15 (D) 20 (E) 25 5 John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? (A) 30 (B) 50 (C) 60 (D) 90 (E) 120 6 Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade. (A) 129 (B) 137 (C) 174 (D) 223 (E) 411 7 Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver’s seat. How many seating arrangements are possible? (A) 4 (B) 12 (C) 16 (D) 24 (E) 48 1 8 The lines x = y + a and y = 4 3 (C) 1 (A) 0 (B) 4 1 x + b intersect at the point (1, 2). What is a + b? 4 9 (D) 2 (E) 4

USA
AMC 12 2006

9 How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order? (A) 21 (B) 34 (C) 51 (D) 72 (E) 150 10 In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? (A) 43 (B) 44 (C) 45 (D) 46 (E) 47 11 Joe and JoAnn each bought 12 ounces of co?ee in a 16-ounce cup. Joe drank 2 ounces of his co?ee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the co?ee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe’s co?ee to that in JoAnn’s co?ee? 6 13 14 7 (A) (B) (C) 1 (D) (E) 7 14 13 6 12 The parabola y = ax2 + bx + c has vertex (p, p) and y -intercept (0, p), where p 6= 0. What is b? (A) p (B) 0 (C) 2 (D) 4 (E) p 13 Rhombus ABCD is similar to rhombus BF DE . The area of rhombus ABCD is 24, and \BAD = 60 . What is the area of rhombus BF DE ? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 613[/img ] p p (A) 6 (B) 4 3 (C) 8 (D) 9 (E) 6 3 14 Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4 cents per glob and J blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is U N IQ982aef 5e5 QIN U 2.53. Assume that B, J, and N are all positive integers with N > 1. What is the cost of the jam Elmo uses to make the sandwiches? (A) U N IQ982aef 5e5 QIN U 1.05 (B) U N IQ982aef 5e5 QIN U 1.25 (C) U N IQ982aef 5e5 QIN U 1.45 (D) U N IQ982aef 5e5 QIN U 1.65 (E) U N IQ982aef 5e5 QIN U 1.85 15 Circles with centers O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B are on the circle centered at O, and points C and D are on the circle centered at P , such that AD and BC are common external tangents to the circles. What is the area of hexagon AOBCP D? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 614[/img ] p p p p (A) 18 3 (B) 24 2 (C) 36 (D) 24 3 (E) 32 2

USA
AMC 12 2006

16 Regular hexagon ABCDEF has vertices A and C at (0, 0) and (7, 1), respectively. What is its area? p p p p (A) 20 3 (B) 22 3 (C) 25 3 (D) 27 3 (E) 50 17 For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio 1 : 2 : 3 : 4 : 5 : 6. What is the probability of rolling a total of 7 on the two dice? 4 1 8 1 2 (A) (B) (C) (D) (E) 63 8 63 6 7 18 An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many di?erent points could be the ?nal point? (A) 120 (B) 121 (C) 221 (D) 230 (E) 231 19 Mr. Jones has eight children of di?erent ages. On a family trip his olderst child, who is 9, spots a license plate with a 4-digit number in which each of two digitrs appears two times. quot;Look, daddy!quot; she exclaims. quot;That number is evenly divisible by the age of each of us kids!quot; quot;That’s right,quot; replies Mr. Jones, quot;and the last two digits just happen to be my age.quot; Which of the following is not the age of one of Mr. Jones’s children? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 20 Let x be chosen at random from the interval (0, 1). What is the probability that blog10 4xc blog10 xc = 0

Here bxc denotes the greatest integer that is less than or equal to x. 1 3 1 1 1 (A) (B) (C) (D) (E) 8 20 6 5 4 21 Rectangle ABCD has area 2006. An ellipse with area 2006? passes through A and C and has foci at B and D. What is the perimeter of the rectangle? (The area of an ellipse is ? ab, where 2a and 2b are the lengths of its axes.) p p p p 16 2006 1003 32 1003 (A) (B) (C) 8 1003 (D) 6 2006 (E) ? 4 ? 22 Suppose a, b, and c are positive integers with a + b + c = 2006, and a!b!c! = m · 10n , where m and n are integers and m is not divisible by 10. What is the smallest possible value of n? (A) 489 (B) 492 (C) 495 (D) 498 (E) 501

USA
AMC 12 2006

23 Isosceles 4ABC has a right angle at C . Point P q is inside 4ABC , such that P A = 11, P B = 7, p and P C = 6. Legs AC and BC have length s = a + b 2, where a and b are positive integers. What is a + b? [img]http://www.artofproblemsolving.com/Forum/albump ic.php?pici d = 615[/img ] (A) 85 (B) 91 (C) 108 (D) 121 (E) 127 ? ? 24 Let S be the set of all points (x, y ) in the coordinate plane such that 0 ? x ? and 0 ? y ? . 2 2 What is the area of the subset of S for which sin2 x (A) ?2 9 (B) ?2 8 (C) ?2 6 (D) sin x sin y + sin2 y ? 3? 2 16 (E) 2? 2 9 an | for n 1. 3 4

25 A sequence a1 , a2 , . . . of non-negative integers is de?ned by the rule an+2 = |an+1 If a1 = 999, a2 < 999, and a2006 = 1, how many di?erent values of a2 are possible? (A) 165 (B) 324 (C) 495 (D) 499 (E) 660

USA
AMC 12 2007

A

1 One ticket to a show costs U N IQ1cbaf 29d6 QIN U 20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? (A) 2 (B) 5 (C) 10 (D) 15 (E) 20 2 An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is ?lled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? (A) 0.5 (A) 4 (B) 1 (B) 8 (C) 1.5 (C) 12 (D) 2 (D) 16 (E) 2.5 (E) 20 3 The larger of two consecutive odd integers is three times the smaller. What is their sum? 4 Kate rode her bicycle for 40 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour? (A) 7 (B) 9 (C) 10 (D) 12 (E) 14 5 Last year Mr. John Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state taxes. He paid a total of U N IQ276823f 77 QIN U 10, 500 for both taxes. How many dollars was the inheritance? (A) 30, 000 (B) 32, 500 (C) 35, 000 (D) 37, 500 (E) 40, 000 6 Triangles ABC and ADC are isosceles with AB = BC and AD = DC. Point D is inside 4ABC, \ABC = 40 , and \ADC = 140 . What is the degree measure of \BAD? (A) 20 (B) 30 (C) 40 (D) 50 (E) 60 7 Let a, b, c, d, and e be ?ve consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 30. Which of the following can be found? (A) a (B) b (C) c (D) d (E) e 8 A star-polygon is drawn on a clock face by drawing a chord from each number to the ?rth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star-polygon? (A) 20 (B) 24 (C) 30 (D) 36 (E) 60

USA
AMC 12 2007

9 Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan’s distance from his home to his distance from the stadium? 2 3 4 5 6 (A) (B) (C) (D) (E) 3 4 5 6 7 10 A triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle of radius 3. What is the area of the triangle? (A) 8.64 (B) 12 (C) 5? (D) 17.28 (E) 18 11 A ?nite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the ?rst term. For example, such a sequence might begin with the terms 247, 275, and 756 and end with the term 824. Let S be the sum of all the terms in the sequence. What is the largest prime factor that always divides S ? (A) 3 (B) 7 (C) 13 (D) 37 (E) 43 12 Integers a, b, c, and d, not necessarily distinct, are chosen independantly and at random from 0 to 2007, inclusive. What is the probability that ad bc is even? 3 7 1 9 5 (A) (B) (C) (D) (E) 8 16 2 16 8 13 A piece of cheese is located at (12, 10) in a coordinate plane. A mouse is at (4, 2) and is running up the line y = 5x + 18. At the point (a, b) the mouse starts getting farther from the cheese rather than closer to it. What is a + b? (A) 6 (B) 10 (C) 14 (D) 18 (E) 22 14 Let a, b, c, d, and e be distinct integers such that (6 What is a + b + c + d + e? (A) 5 (B) 17 (C) 25 (D) 27 (E) 30 15 The set {3, 6, 9, 10} is augmented by a ?fth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n? (A) 7 (B) 9 (C) 19 (D) 24 (E) 26 a)(6 b)(6 c)(6 d)(6 e) = 45.

USA
AMC 12 2007

16 How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? (C) 112 (D) 120 (E) 256 r 5 17 Suppose that sin a + sin b = and cos a + cos b = 1. What is cos(a 3 r 5 1 1 2 (A) 1 (B) (C) (D) (E) 1 3 3 2 3 (A) 96 (B) 104

b)?

18 The polynomial f (x) = x4 + ax3 + bx2 + cx + d has real coe cients, and f (2i) = f (2 + i) = 0. What is a + b + c + d? (A) 0 (B) 1 (C) 4 (D) 9 (E) 16 19 Triangles ABC and ADE have areas 2007 and 7002, respectively, with B = (0, 0), C = (223, 0), D = (680, 380), and E = (689, 389). What is the sum of all possible x-coordinates of A? (A) 282 (B) 300 (C) 600 (D) 900 (E) 1200 20 Corners are sliced o? a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? p p p p p 5 2 7 10 7 2 3 2 2 8 2 11 6 4 2 (A) (B) (C) (D) (E) 3 3 3 3 3 21 The sum of the zeros, the product of the zeros, and the sum of the coe cients of the function f (x) = ax2 + bx + c are equal. Their common value must also be which of the following?

(A) the coe cient of x2 (B) the coe cient of x (C) the y - intercept of the graph of y = f (x) (D) one of the x - intercepts of the graph of y = f (x) (E) the mean of the x - intercepts of the graph of f (x) 22 For each positive integer n, let S (n) denote the sum of the digits of n. For how many values of n is n + S (n) + S (S (n)) = 2007? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 23 Square ABCD has area 36, and AB is parallel to the x-axis. Vertices A, B, and C are on the graphs of y = loga x, y = 2 loga x, and y = 3 loga x, respectively. What is a? p p p p 6 3 (A) 3 (B) 3 (C) 6 (D) 6 (E) 6 24 For each integer n > 1, let F (n) be the number of solutions of the equation sin x = sin nx on 2007 X the interval [0, ? ]. What is F (n)?
n=2

USA
AMC 12 2007

(A) 2, 014, 524

(B) 2, 015, 028

(C) 2, 015, 033

(D) 2, 016, 532

(E) 2, 017, 033

25 Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of {1, 2, 3, . . . , 12}, including the empty set, are spacy? (A) 121 (B) 123 (C) 125 (D) 127 (E) 129

USA
AMC 12 2007

B

1 Isabella’s house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted? (A) 678 (B) 768 (C) 786 (D) 867 (E) 876 2 A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents’ SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? (A) 22 (B) 24 (C) 25 (D) 26 (E) 28 3 The point O is the center of the circle circumscribed about 4ABC , with \BOC = 120 and \AOB = 140 , as shown. What is the degree measure of \ABC ? [asy]unitsize(3cm); defaultpen(fontsize(8pt)); pair B=dir(80); pair C=dir(-40); pair A=dir(220); pair O=(0,0); draw(Circle(O,1)); draw(O–C–A–cycle); draw(O–B–A); draw(O–B–C); dot(O); dot(A); dot(B); dot(C); 4 At Frank’s Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas? (A) 6 (B) 8 (C) 9 (D) 12 (E) 18 5 The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the ?rst 22 and leave the last three unanswered. How many of the ?rst 22 problems must she solve correctly in order to score at least 100 points? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17 6 Triangle ABC has side lengths AB = 5, BC = 6, and AC = 7. Two bugs start simultaneously from A and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point D. What is BD? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

label(quot;36;O36;quot;,O,S); label(quot;36;120 36; quot; , O, N E ); label(quot; 36; 140 36; quot; , O, N W ); labe

USA
AMC 12 2007

7 All sides of the convex pentagon ABCDE are of equal length, and \A = \B = 90 . What is the degree measure of \E ? (A) 90 (B) 108 (C) 120 (D) 144 (E) 150 8 Tom’s age is T years, which is also the sum of the ages of his three children. His age N years T ago was twice the sum of their ages then. What is ? N (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 9 A function f has the property that f (3x f (5)? (A) 7 (B) 13 (C) 31 (D) 111 1) = x2 + x + 1 for all real numbers x. What is (E) 211

10 Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30% of the group are girls. How many girls were initially in the group? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12 11 The angles of quadrilateral ABCD satisfy \A = 2\B = 3\C = 4\D. What is the degree measure of \A, rounded to the nearest whole number? (A) 125 (B) 144 (C) 153 (D) 173 (E) 180 12 A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the test was 84. The juniors all received the same score, and the average score of the seniors was 83. What score did each of the juniors receive on the test? (A) 85 (B) 88 (C) 93 (D) 94 (E) 98 13 A tra c light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? 1 1 1 1 1 (A) (B) (C) (D) (E) 63 21 10 7 3 14 Point P is inside equilateral 4ABC . Points Q, R and S are the feet of the perpendiculars from P to AB, BC , and CA, respectively. Given that P Q = 1, P R = 2, and P S = 3, what is AB ? p p (A) 4 (B) 3 3 (C) 6 (D) 4 3 (E) 9 15 The geometric series a + ar + ar2 + ... has a sum of 7, and the terms involving odd powers of r have a sum of 3. What is a + r? 4 12 3 7 5 (A) (B) (C) (D) (E) 3 7 2 3 2

USA
AMC 12 2007

16 Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? (A) 15 (B) 18 (C) 27 (D) 54 (E) 81 17 If a is a nonzero integer and b is a positive number such that ab2 = log10 b, what is the median of the set {0, 1, a, b, 1/b}? 1 (A) 0 (B) 1 (C) a (D) b (E) b 18 Let a, b, and c be digits with a 6= 0. The three-digit integer abc lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer acb lies two thirds of the way between the same two squares. What is a + b + c? (A) 10 (B) 13 (C) 16 (D) 18 (E) 21 19 Rhombus ABCD, with a side length 6, is rolled to form a cylinder of volume 6 by taping AB to DC. What is sin(\ABC )? p ? 1 ? ? 3 (A) (B) (C) (D) (E) 9 2 6 4 2 20 The parallelogram bounded by the lines y = ax + c, y = ax + d, y = bx + c and y = bx + d has area 18. The parallelogram bounded by the lines y = ax + c, y = ax d, y = bx + c, and y = bx d has area 72. Given that a, b, c, and d are positive integers, what is the smallest possible value of a + b + c + d? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17 21 The ?rst 2007 positive integers are each written in base 3. How many of these base-3 representations are palindromes? (A palindrome is a number that reads the same forward and backward.) (A) 100 (B) 101 (C) 102 (D) 103 (E) 104 22 Two particles move along the edges of equilateral triangle 4ABC in the direction A!B!C!A starting simultaneously and moving at the same speed. One starts at A, and the other starts at the midpoint of BC . The midpoint of the line segment joining the two particles traces out a path that encloses a region R. What is the ratio of the area of R to the area of 4ABC ? 1 1 1 1 1 (A) (B) (C) (D) (E) 16 12 9 6 4

USA
AMC 12 2007

23 How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 times their perimeters? (A) 6 (B) 7 (C) 8 (D) 10 (E) 12 24 How many pairs of positive integers (a, b) are there such that gcd(a, b) = 1 and a 14b + b 9a is an integer? (A) 4 (B) 6 (C) 9 (D) 12 (E) in?nitely many 25 Points A, B , C , D, and E are located in 3-dimensional space with AB = BC = CD = DE = EA = 2 and \ABC = \CDE = \DEA = 90 . The plane of 4ABC is parallel to DE . What is the area of 4BDE ? p p p p (A) 2 (B) 3 (C) 2 (D) 5 (E) 6

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 1 2 + ? 2 3 5 (C) (D) 3 3

2 What is the reciprocal of (A) 6 7 (B) 7 6

(E)

7 2

2 3 Suppose that of 10 bananas are worth as much as 8 oranges. How many oranges are worth 3 1 as much is of 5 bananas? 2 5 7 (A) 2 (B) (C) 3 (D) (E) 4 2 2 4 Which of the following is equal to the product 8 12 16 4n + 4 2008 · · ··· ··· ? 4 8 12 4n 2004 (A) 251 (B) 502 (C) 1004 (D) 2008

(E) 4016

5 Suppose that 2x x 3 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 di?erent stores. Store A o?ers 15% o? the sticker price followed by a U N IQed22a9d61 QIN U 90 rebate, and store B o?ers 25% o? the same sticker price with no rebate. Heather saves U N IQed22a9d61 QIN U 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

USA
AMC 12 2008

7 While Steve and LeRoy are ?shing 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? p p (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); ?lldraw((0,0)–(21.6,0)–(21.6,2.7)–(0,2.7)–cycle,grey,black); ?lldraw((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 satis?ed by t? ? ◆ ? ◆ ? ◆ 1 1 1 1 1 1 (A) + (t + 1) = 1 (B) + t+1=1 (C) + t=1 5 7 5 7 ?5 7◆ 1 1 (D) + (t 1) = 1 (E) (5 + 7)t = 1 5 7 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); label(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

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 de?ned by g (x) = 1 f (x +1)? (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 \AOB = 60 . A second circle is internally tangent to the ?rst and tangent to both OA and OB . What is the ratio of the area of the smaller circle to that of the larger circle? 1 1 1 1 1 (A) (B) (C) (D) (E) 16 9 8 6 4 14 What is the area of the region de?ned by the inequality |3x 7 9 (A) 3 (B) (C) 4 (D) (E) 5 2 2 15 Let k = 20082 + 22008 . What is the units digit of k 2 + 2k ? (A) 0 (B) 2 (C) 4 (D) 6 (E) 8 16 The numbers log(a3 b7 ), log(a5 b12 ), and log(a8 b15 ) are the ?rst 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 an = an 1 /2 if an 1 is even and an = 3an 1 + 1 if an 1 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 ? p p p p (A) 85 (B) 90 (C) 95 (D) 10 (E) 105 19 In the expansion of what is the coe cient of x28 ? (A) 195 (B) 196 1 + x + x2 + · · · + x27 1 + x + x2 + · · · + x14 , (C) 224 (D) 378 (E) 405
2

18| + |2y + 7| ? 3?

20 Triangle ABC has AC = 3, BC = 4, and AB = 5. 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 ?

USA
AMC 12 2008

21 A permutation (a1 , a2 , a3 , a4 , a5 ) of (1, 2, 3, 4, 5) is heavy-tailed if a1 + a2 < a4 + a5 . 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); p label(quot;36;x36;quot;,(-2.687,0),E); label(quot;36;136;quot;,(-3.187,1.5513),S);[/asy] (A) 2 5 p p p p p 3 7 3 5+2 3 3 (B) 3 (C) (D) 2 3 (E) 2 2 23 The solutions of the equation z 4 + 4z 3 i 6z 2 4zi i = 0 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 \C = 60 and BC = 4. Point D is the midpoint of BC . What is the largest possible value of tan \BAD? p p p p 3 3 3 3 (A) (B) (C) p (D) p (E) 1 6 3 2 2 4 2 3 25 A sequence (a1 , b1 ), (a2 , b2 ), (a3 , b3 ), . . . of points in the coordinate plane satis?es p p (an+1 , bn+1 ) = ( 3an bn , 3bn + an ) for n = 1, 2, 3, . . .. Suppose that (a100 , b100 ) = (2, 4). What is a1 + b1 ? 1 1 1 (A) (B) (C) 0 (D) 98 97 99 2 2 2 (E) 1 296

1 ? 10 28 ? 3 (E) 10 28 (A)

p ? 2 p ? 2

(B)

3 ? 10 56

p ? 2

(C)

1 ? 10 14

p ? 2

(D)

5 ? 10 56

p ? 2

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 di?erent 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 di?erence between the two diagonal sums?
1 8 2 3 4 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 U N IQ3d9e3974f QIN U 15, 000, and that the total of all players’ salaries for each team cannot exceed U N IQ3d9e3974f QIN U 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 , \AOC = 30 , and \DOB = 45 . 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), W N W ); label(quot; 36; 30 36; quot; , (0.5, 0.1), EN E ); draw(A B ); draw(O D); draw(O C ); [/asy ] 2 1 5 7 3 (A) (B) (C) (D) (E) 9 4 18 24 10 5 A class collects U N IQcadf 21aaa QIN U 50 to buy ?owers for a classmate who is in the hospital. Roses cost U N IQcadf 21aaa QIN U 3 each, and carnations cost U N IQcadf 21aaa QIN U 2 each.

USA
AMC 12 2008

No other ?owers are to be used. How many di?erent bouquets could be purchased for exactly U N IQcadf 21aaa QIN U 50? (A) 1 (B) 7 (C) 9 (D) 16 (E) 17 6 Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then ?ips 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 ?ips 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, de?ne aU N IQ8dc89894c QIN U b = (a b)2 . What is (x y )2 U N IQ8dc89894c QIN U (y x)2 ? (A) 0 (B) x2 + y 2 (C) 2x2 (D) 2y 2 (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? 1 1 1 5 1 (A) (B) (C) (D) (E) 36 13 10 36 5 9 Points A and B are on a circle of radius 5 and AB = 6. Point C is the midpoint of the minor arc AB . What is the length of the line segment AC ? p p p 7 (A) 10 (B) (C) 14 (D) 15 (E) 4 2 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 1 11 A cone-shaped mountain has its base on the ocean ?oor and has a height of 8000 feet. The top 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? p (A) 4000 (B) 2000(4 2) (C) 6000 (D) 6400 (E) 7000 12 For each positive integer n, the mean of the ?rst 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

USA
AMC 12 2008

13 Vertex E of equilateral 4ABE is in the interior of unit square ABCD. Let R be the region 1 consisting of all points inside ABCD and outside 4ABE whose distance from AD is between 3 2 and . What is the area of R? 3 p p p p p 12 5 3 3 3 3 3 12 5 3 (B) (C) (D) (E) (A) 72 36 18 9 12 14 A circle has a radius of log10 (a2 ) and a circumference of log10 (b4 ). What is loga b? 1 1 (A) (B) (C) ? (D) 2? (E) 102? 4? ? 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? p p p 1 2 (A) (B) (C) 1 (D) 3 (E) 2 3 4 4 16 A rectangular ?oor measures a by b feet, where a and b are positive integers with b > a. An artist paints a rectangle on the ?oor with the sides of the rectangle parallel to the sides of the ?oor. The unpainted part of the ?oor forms a border of width 1 foot around the painted rectangle and occupies half of the area of the entire ?oor. 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 y = x2 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? p p p (A) 392 (B) 196 6 (C) 392 2 (D) 392 3 (E) 784 19 A function f is de?ned by f (z ) = (4 + i)z 2 + ?z + for all complex numbers z , where ? and are complex numbers and i2 = 1. Suppose that f (1) and f (i) are both real. What is the smallest possible value of |?| + | | p p (A) 1 (B) 2 (C) 2 (D) 2 2 (E) 4

USA
AMC 12 2008

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 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 ?rst choice, from the line segment joining (0, 1) to (2, 1). What is the probability that circles A and B intersect? p p p p p 2+ 2 3 3+2 2 2 1 2+ 3 4 3 3 (A) (B) (C) (D) (E) 4 8 2 4 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? 11 4 81 3 17 (A) (B) (C) (D) (E) 20 7 140 5 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 p the graph of y = x. For every positive integer n, An 1 Bn An is an equilateral triangle. What is the least n for which the length A0 An 100? (A) 13 (B) 15 (C) 17 (D) 19 (E) 21 25 Let ABCD be a trapezoid with AB ||CD, AB = 11, BC = 5, CD = 19, and DA = 7. Bisectors of \A and \D meet at P , and bisectors of \B and \C meet at Q. What is the area of hexagon ABQCDP ? p p p p p (A) 28 3 (B) 30 3 (C) 32 3 (D) 35 3 (E) 36 3


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