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2002 AMC 12B problems

All 25 problems from the 2002 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 1The 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 Theory
  2. 2Problem 2What is the value of (3x-2)(4x+1) -(3x-2)4x+1 when x=4?Algebra
  3. 3Problem 3For how many positive integers n is n^2-3n+2 a prime number?Algebra
  4. 4Problem 4Let n be a positive integer such that dfrac 12+dfrac 13+dfrac 17+dfrac 1n is an integer. Which of the following statements is not true:Algebra
  5. 5Problem 5Let v, w, x, y, and z be the degree measures of the five angles of a pentagon. Suppose v< w< x< y< z and v, w, x, y, z form an arithmetic sequence…Algebra
  6. 6Problem 6Suppose that a and b are nonzero real numbers, and that the equation x^2+ax+b=0 has solutions a and b. Then the pair (a,b) isAlgebra
  7. 7Problem 7The product of three consecutive positive integers is 8 times their sum. What is the sum of their squares?Algebra
  8. 8Problem 8Suppose July of year N has five Mondays. Which of the following must occur five times in August of year N? (Note: Both months have 31 days.)Algebra
  9. 9Problem 9If a, b, c, d are positive real numbers such that a, b, c, d form an increasing arithmetic sequence and a, b, d form a geometric sequence, then a/d isAlgebra
  10. 10Problem 10How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19}?Number Theory
  11. 11Problem 11The positive integers A, B, A-B, and A+B are all prime numbers. The sum of these four primes isNumber Theory
  12. 12Problem 12For how many integers n is n/20-n the square of an integer?Number Theory
  13. 13Problem 13The sum of 18 consecutive positive integers is a perfect square. The smallest possible value of this sum isAlgebra
  14. 14Problem 14Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?Counting & Probability
  15. 15Problem 15How many four-digit numbers N have the property that the three-digit number obtained by removing the leftmost digit is one ninth of N?Number Theory
  16. 16Problem 16Juan 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…Counting & Probability
  17. 17Problem 17Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower…Algebra
  18. 18Problem 18A 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…Geometry
  19. 19Problem 19If 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 isAlgebra
  20. 20Problem 20Let △ XOY 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 YM=22, find…Geometry
  21. 21Problem 21For all positive integers n less than 2002, let a_n= 11, if n is divisible by 13 and 14; 13, if n is divisible by 14 and 11; 14, if n is divisible by…Number Theory
  22. 22Problem 22For all integers n greater than 1, define a_n=1/log _n 2002. Let b=a_2+a_3+a_4+a_5 and c=a_10+a_11+a_12+a_13+a_14. Then b-c equalsAlgebra
  23. 23Problem 23In △ ABC, we have AB=1 and AC=2. Side BC and the median from A to BC have the same length. What is BC?Geometry
  24. 24Problem 24A convex quadrilateral ABCD with area 2002 contains a point P in its interior such that PA=24, PB=32, PC=28, and PD=45. Find the perimeter of ABCD.Geometry
  25. 25Problem 25Let f(x)=x^2+6x+1, and let R denote the set of points (x,y) in the coordinate plane such that f(x)+f(y)le 0 and f(x)-f(y)le 0. The area of R is…Geometry

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