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2004 AMC 12B

All 25 problems from the 2004 AMC 12B. Take it as a timed mock test — 1:15:00 on the clock, scored the way the real contest is — or read through the problems and open any of them for hints and a worked solution.

25 problems · 6 points correct, 1.5 for blank, 0 for wrong

  1. Problem 1

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    At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 4848 free throws. How many free throws did she make at the first practice?

  2. Problem 2

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    In the expression c⋅ab−d,c \cdot a^b - d, the values of a,a, b,b, c,c, and dd are 0,0, 1,1, 2,2, and 3,3, although not necessarily in that order. What is the maximum possible value of the result?

  3. Problem 3

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    If xx and yy are positive integers for which 2x3y=1296,2^x 3^y = 1296, what is the value of x+y?x + y?

  4. Problem 4

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    An integer x,x, with 10≤x≤99,10 \le x \le 99, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of xx is a 7?7?

  5. Problem 5

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    On a trip from the United States to Canada, Isabella took dd U.S. dollars. At the border she exchanged them all, receiving 1010 Canadian dollars for every 77 U.S. dollars. After spending 6060 Canadian dollars, she had dd Canadian dollars left. What is the sum of the digits of d?d?

  6. Problem 6

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    Minneapolis-St. Paul International Airport is 88 miles southwest of downtown St. Paul and 1010 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?

  7. Problem 7

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    A square has sides of length 10,10, and a circle centered at one of its vertices has radius 10.10. What is the area of the union of the regions enclosed by the square and the circle?

  8. Problem 8

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    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 100100 cans, how many rows does it contain?

  9. Problem 9

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    The point (−3,2)(-3, 2) is rotated 90∘90^\circ clockwise around the origin to point B.B. Point BB is then reflected in the line y=xy = x to point C.C. What are the coordinates of C?C?

  10. Problem 10

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    An annulus is the region between two concentric circles. The concentric circles in the figure have radii bb and c,c, with b>c.b \gt c. Let OX‾\overline{OX} be a radius of the larger circle, let XZ‾\overline{XZ} be tangent to the smaller circle at Z,Z, and let OY‾\overline{OY} be the radius of the larger circle that contains Z.Z. Let a=XZ,a = XZ, d=YZ,d = YZ, and e=XY.e = XY. What is the area of the annulus?

  11. Problem 11

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    All the students in an algebra class took a 100100-point test. Five students scored 100,100, each student scored at least 60,60, and the mean score was 76.76. What is the smallest possible number of students in the class?

  12. Problem 12

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    In the sequence 2001,2001, 2002,2002, 2003,2003, …,\ldots, 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=2000.2001 + 2002 - 2003 = 2000. What is the 20042004th term in this sequence?

  13. Problem 13

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    If f(x)=ax+bf(x) = ax + b and f−1(x)=bx+af^{-1}(x) = bx + a with aa and bb real, what is the value of a+b?a + b?

  14. Problem 14

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    In △ABC,\triangle ABC, AB=13,AB = 13, AC=5AC = 5 and BC=12.BC = 12. Points MM and NN lie on AC‾\overline{AC} and BC‾,\overline{BC}, respectively, with CM=CN=4.CM = CN = 4. Points JJ and KK are on AB‾\overline{AB} so that MJ‾\overline{MJ} and NK‾\overline{NK} are perpendicular to AB‾.\overline{AB}. What is the area of pentagon CMJKN?CMJKN?

  15. Problem 15

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    The two digits in Jack’s age are the same as the digits in Bill’s age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?

  16. Problem 16

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    A function ff is defined by f(z)=iz‾,f(z) = i\overline{z}, where i=−1i = \sqrt{-1} and z‾\overline{z} is the complex conjugate of z.z. How many values of zz satisfy both ∣z∣=5|z| = 5 and f(z)=z?f(z) = z?

  17. Problem 17

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    For some real numbers aa and b,b, the equation 8x3+4ax2+2bx+a=08x^3 + 4ax^2 + 2bx + a = 0 has three distinct positive roots. If the sum of the base-22 logarithms of the roots is 5,5, what is the value of a?a?

  18. Problem 18

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    Points AA and BB are on the parabola y=4x2+7x−1,y = 4x^2 + 7x - 1, and the origin is the midpoint of AB‾.\overline{AB}. What is the length of AB?AB?

  19. Problem 19

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    A truncated cone has horizontal bases with radii 1818 and 2.2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?

  20. Problem 20

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    Each face of a cube is painted either red or blue, each with probability 12.\tfrac12. 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?

  21. Problem 21

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    The graph of 2x2+xy+3y22x^2 + xy + 3y^2 −11x−20y+40=0- 11x - 20y + 40 = 0 is an ellipse in the first quadrant of the xyxy-plane. Let aa and bb be the maximum and minimum values of yx\dfrac{y}{x} over all points (x,y)(x, y) on the ellipse. What is the value of a+b?a + b?

  22. Problem 22

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    The square 50bcdefgh2\begin{array}{|c|c|c|} \hline 50 & b & c \\ \hline d & e & f \\ \hline g & h & 2 \\ \hline \end{array} 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?g?

  23. Problem 23

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    The polynomial x3−2004x2+mx+nx^3 - 2004x^2 + mx + n has integer coefficients 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 nn are possible?

  24. Problem 24

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    In △ABC,\triangle ABC, AB=BC,AB = BC, and BD‾\overline{BD} is an altitude. Point EE is on the extension of AC‾\overline{AC} such that BE=10.BE = 10. The values of tan⁡∠CBE,\tan \angle CBE, tan⁡∠DBE,\tan \angle DBE, and tan⁡∠ABE\tan \angle ABE form a geometric progression, and the values of cot⁡∠DBE,\cot \angle DBE, cot⁡∠CBE,\cot \angle CBE, cot⁡∠DBC\cot \angle DBC form an arithmetic progression. What is the area of △ABC?\triangle ABC?

  25. Problem 25

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    Given that 220042^{2004} is a 604604-digit number whose first digit is 1,1, how many elements of the set S={20,21,22,…,22003}S = \{2^0, 2^1, 2^2, \ldots, 2^{2003}\} have a first digit of 4?4?

0 of 25 answered. Each problem left blank still scores 1.5 points.

Practise one problem at a time, with hints and solutions, on the practice page. Or work through the same ideas across every year by topic, or browse the full AMC 12 archive.