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2004 AMC 12B problems

All 25 problems from the 2004 AMC 12B, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1At 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 48…Algebra
  2. 2Problem 2In the expression c · a^b - 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…Algebra
  3. 3Problem 3If x and y are positive integers for which 2^x 3^y = 1296, what is the value of x + y?Algebra
  4. 4Problem 4An 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?Counting & Probability
  5. 5Problem 5On 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…Algebra
  6. 6Problem 6Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the…Geometry
  7. 7Problem 7A 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…Geometry
  8. 8Problem 8A 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…Algebra
  9. 9Problem 9The point (-3, 2) is rotated 90° clockwise around the origin to point B. Point B is then reflected in the line y = x to point C. What are the…Geometry
  10. 10Problem 10An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c, with b > c. Let OX be a radius of…Geometry
  11. 11Problem 11All 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…Algebra
  12. 12Problem 12In 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…Algebra
  13. 13Problem 13If f(x) = ax + b and f^-1(x) = bx + a with a and b real, what is the value of a + b?Algebra
  14. 14Problem 14In △ ABC, 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 MJ and NK…Geometry
  15. 15Problem 15The 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…Algebra
  16. 16Problem 16A function f is defined 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?Algebra
  17. 17Problem 17For some real numbers a and b, the equation 8x^3 + 4ax^2 + 2bx + a = 0 has three distinct positive roots. If the sum of the base-2 logarithms of the…Algebra
  18. 18Problem 18Points A and B are on the parabola y = 4x^2 + 7x - 1, and the origin is the midpoint of AB. What is the length of AB?Geometry
  19. 19Problem 19A 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…Geometry
  20. 20Problem 20Each face of a cube is painted either red or blue, each with probability tfrac 12. The color of each face is determined independently. What is the…Geometry
  21. 21Problem 21The graph of 2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0 is an ellipse in the first quadrant of the xy-plane. Let a and b be the maximum and minimum values…Geometry
  22. 22Problem 22The square |c|c|c| hline 50 b c hline d e f hline g h 2 hline is a multiplicative magic square. That is, the product of the numbers in each row…Algebra
  23. 23Problem 23The polynomial x^3 - 2004x^2 + mx + n has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the…Algebra
  24. 24Problem 24In △ ABC, 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…Geometry
  25. 25Problem 25Given that 2^2004 is a 604-digit number whose first digit is 1, how many elements of the set S = {2^0, 2^1, 2^2, …, 2^2003} have a first digit of 4?Number Theory

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.