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2005 AMC 12A

All 25 problems from the 2005 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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    Two is 10%10\% of xx and 20%20\% of y.y. What is x−y?x - y?

  2. Problem 2

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    The equations 2x+7=32x + 7 = 3 and bx−10=−2bx - 10 = -2 have the same solution for x.x. What is the value of b?b?

  3. Problem 3

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    A rectangle with a diagonal of length xx is twice as long as it is wide. What is the area of the rectangle?

  4. Problem 4

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    A store normally sells windows at $100\$100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

  5. Problem 5

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    The average (mean) of 2020 numbers is 30,30, and the average of 3030 other numbers is 20.20. What is the average of all 5050 numbers?

  6. Problem 6

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    Josh and Mike live 1313 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-fifths of Mike’s rate. How many miles had Mike ridden when they met?

  7. Problem 7

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    Square EFGHEFGH is inside square ABCDABCD so that each side of EFGHEFGH can be extended to pass through a vertex of ABCD.ABCD. Square ABCDABCD has side length 50,\sqrt{50}, EE is between BB and H,H, and BE=1.BE = 1. What is the area of the inner square EFGH?EFGH?

  8. Problem 8

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    Let A,A, M,M, and CC be digits with (100A+10M+C)⋅(A+M+C)=2005. \begin{aligned} &(100A + 10M + C) \\ &\quad {}\cdot (A + M + C) = 2005. \end{aligned} What is A?A?

  9. Problem 9

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    There are two values of aa for which the equation 4x2+ax+8x+9=04x^2 + ax + 8x + 9 = 0 has only one solution for x.x. What is the sum of those values of a?a?

  10. Problem 10

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    A wooden cube nn units on a side is painted red on all six faces and then cut into n3n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n?n?

  11. Problem 11

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    How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

  12. Problem 12

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    A line passes through A(1,1)A(1, 1) and B(100,1000).B(100, 1000). How many other points with integer coordinates are on the line and strictly between AA and B?B?

  13. Problem 13

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    In the five-sided star shown, the letters A,A, B,B, C,C, D,D, and EE are replaced by the numbers 3,3, 5,5, 6,6, 7,7, and 9,9, although not necessarily in that order. The sums of the numbers at the ends of the line segments AB,AB, BC,BC, CD,CD, DE,DE, and EAEA form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

  14. Problem 14

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    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?

  15. Problem 15

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    Let ABAB be a diameter of a circle and CC be a point on ABAB with 2⋅AC=BC.2 \cdot AC = BC. Let DD and EE be points on the circle such that DC⊥ABDC \perp AB and DEDE is a second diameter. What is the ratio of the area of △DCE\triangle DCE to the area of △ABD?\triangle ABD?

  16. Problem 16

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    Three circles of radius ss are drawn in the first quadrant of the xyxy-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the xx-axis, and the third is tangent to the first circle and the yy-axis. A circle of radius r>sr \gt s is tangent to both axes and to the second and third circles. What is rs?\frac{r}{s}?

  17. Problem 17

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    A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1.1. The cube is then cut in the same manner along the dashed lines shown in Figure 2.2. This creates nine pieces. What is the volume of the piece that contains vertex W?W?

  18. Problem 18

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    Call a number “prime-looking” if it is composite but not divisible by 2,2, 3,3, or 5.5. The three smallest prime-looking numbers are 49,49, 77,77, and 91.91. There are 168168 prime numbers less than 1000.1000. How many prime-looking numbers are there less than 1000?1000?

  19. Problem 19

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    A faulty car odometer proceeds from digit 33 to digit 5,5, always skipping the digit 4,4, regardless of position. For example, after traveling one mile the odometer changed from 000039000039 to 000050.000050. If the odometer now reads 002005,002005, how many miles has the car actually traveled?

  20. Problem 20

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    For each xx in [0,1],[0, 1], define f(x)={2x,0≤x≤12,2−2x,12<x≤1. f(x) = \begin{cases} 2x, & 0 \le x \le \tfrac{1}{2},\\ 2 - 2x, & \tfrac{1}{2} \lt x \le 1. \end{cases} Let f[2](x)=f(f(x)),f^{[2]}(x) = f(f(x)), and f[n+1](x)=f[n](f(x))f^{[n+1]}(x) = f^{[n]}(f(x)) for each integer n≥2.n \ge 2. For how many values of xx in [0,1][0, 1] is f[2005](x)=12?f^{[2005]}(x) = \tfrac{1}{2}?

  21. Problem 21

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    How many ordered triples of integers (a,b,c),(a, b, c), with a≥2,a \ge 2, b≥1,b \ge 1, and c≥0,c \ge 0, satisfy both log⁡ab=c2005\log_a b = c^{2005} and a+b+c=2005?a + b + c = 2005?

  22. Problem 22

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    A rectangular box PP is inscribed in a sphere of radius r.r. The surface area of PP is 384,384, and the sum of the lengths of its 1212 edges is 112.112. What is r?r?

  23. Problem 23

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    Two distinct numbers aa and bb are chosen randomly from the set {2,22,23,…,225}.\{2, 2^2, 2^3, \ldots, 2^{25}\}. What is the probability that log⁡ab\log_a b is an integer?

  24. Problem 24

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    Let P(x)=(x−1)(x−2)(x−3).P(x) = (x - 1)(x - 2)(x - 3). For how many polynomials Q(x)Q(x) does there exist a polynomial R(x)R(x) of degree 33 such that P(Q(x))=P(x)⋅R(x)?P(Q(x)) = P(x) \cdot R(x)?

  25. Problem 25

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    Let SS be the set of all points with coordinates (x,y,z),(x, y, z), where x,x, y,y, and zz are each chosen from the set {0,1,2}.\{0, 1, 2\}. How many equilateral triangles have all their vertices in S?S?

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.