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2002 AMC 12A

All 25 problems from the 2002 AMC 12A. 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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    Compute the sum of all the roots of (2x+3)(x−4)+(2x+3)(x−6)=0. \begin{aligned} &(2x+3)(x-4) \\ &\quad {}+(2x+3)(x-6)=0. \end{aligned}

  2. Problem 2

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    Cindy was asked by her teacher to subtract 33 from a certain number and then divide the result by 9.9. Instead, she subtracted 99 and then divided the result by 3,3, giving an answer of 43.43. What would her answer have been had she worked the problem correctly?

  3. Problem 3

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    According to the standard convention for exponentiation, 2222=2(2(22))=216=65,536.2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65{,}536. If the order in which the exponentiations are performed is changed, how many other values are possible?

  4. Problem 4

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    Find the degree measure of an angle whose complement is 25%25\% of its supplement.

  5. Problem 5

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    Each of the small circles in the figure 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.

  6. Problem 6

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    For how many positive integers mm does there exist at least one positive integer nn such that m⋅n≤m+n?m \cdot n \le m + n?

  7. Problem 7

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    If an arc of 45∘45^\circ on circle AA has the same length as an arc of 30∘30^\circ on circle B,B, then the ratio of the area of circle AA to the area of circle BB is

  8. Problem 8

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    Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let BB be the total area of the blue triangles, WW the total area of the white squares, and RR the area of the red square. Which of the following is correct?

  9. Problem 9

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    Jamal wants to store 3030 computer files on floppy disks, each of which has a capacity of 1.441.44 megabytes (mb). Three of his files require 0.80.8 mb of memory each, 1212 more require 0.70.7 mb each, and the remaining 1515 require 0.40.4 mb each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

  10. Problem 10

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    Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

  11. Problem 11

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    Mr. Earl E. Bird leaves his house for work at exactly 8:008{:}00 A.M. every morning. When he averages 4040 miles per hour, he arrives at his workplace three minutes late. When he averages 6060 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?

  12. Problem 12

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    Both roots of the quadratic equation x2−63x+k=0x^2 - 63x + k = 0 are prime numbers. The number of possible values of kk is

  13. Problem 13

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    Two different positive numbers aa and bb each differ from their reciprocals by 1.1. What is a+b?a + b?

  14. Problem 14

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    For all positive integers n,n, let f(n)=log⁡2002n2.f(n) = \log_{2002} n^2. Let N=f(11)+f(13)+f(14).N = f(11) + f(13) + f(14). Which of the following relations is true?

  15. Problem 15

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    The mean, median, unique mode, and range of a collection of eight integers are all equal to 8.8. The largest integer that can be an element of this collection is

  16. Problem 16

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    Tina randomly selects two distinct numbers from the set {1,2,3,4,5},\{1, 2, 3, 4, 5\}, and Sergio randomly selects a number from the set {1,2,…,10}.\{1, 2, \ldots, 10\}. The probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina is

  17. Problem 17

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    Several sets of prime numbers, such as {7,83,421,659},\{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?

  18. Problem 18

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    Let C1C_1 and C2C_2 be circles defined by (x−10)2+y2=36(x - 10)^2 + y^2 = 36 and (x+15)2+y2=81,(x + 15)^2 + y^2 = 81, respectively. What is the length of the shortest line segment PQ‾\overline{PQ} that is tangent to C1C_1 at PP and to C2C_2 at Q?Q?

  19. Problem 19

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    The graph of the function ff is shown below. How many solutions does the equation f(f(x))=6f(f(x)) = 6 have?

  20. Problem 20

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    Suppose that aa and bb are digits, not both nine and not both zero, and the repeating decimal 0.ab‾0.\overline{ab} is expressed as a fraction in lowest terms. How many different denominators are possible?

  21. Problem 21

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    Consider the sequence of numbers 4,4, 7,7, 1,1, 8,8, 9,9, 7,7, 6,6, …\ldots For n>2,n \gt 2, the nnth term of the sequence is the units digit of the sum of the two previous terms. Let SnS_n denote the sum of the first nn terms of this sequence. The smallest value of nn for which Sn>10,000S_n \gt 10{,}000 is

  22. Problem 22

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    Triangle ABCABC is a right triangle with ∠ACB\angle ACB as its right angle, m∠ABC=60∘,m\angle ABC = 60^\circ, and AB=10.AB = 10. Let PP be randomly chosen inside △ABC,\triangle ABC, and extend BP‾\overline{BP} to meet AC‾\overline{AC} at D.D. What is the probability that BD>52?BD \gt 5\sqrt{2}?

  23. Problem 23

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    In triangle ABC,ABC, side AC‾\overline{AC} and the perpendicular bisector of BC‾\overline{BC} meet in point D,D, and BD‾\overline{BD} bisects ∠ABC.\angle ABC. If AD=9AD = 9 and DC=7,DC = 7, what is the area of triangle ABD?ABD?

  24. Problem 24

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    Find the number of ordered pairs of real numbers (a,b)(a, b) such that (a+bi)2002=a−bi.(a + bi)^{2002} = a - bi.

  25. Problem 25

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    The nonzero coefficients of a polynomial PP with real coefficients are all replaced by their mean to form a polynomial Q.Q. Which of the following could be a graph of y=P(x)y = P(x) and y=Q(x)y = Q(x) over the interval −4≤x≤4?-4 \le x \le 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.