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2005 AMC 12B

All 25 problems from the 2005 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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    A scout troop buys 10001000 candy bars at a price of five for $2.\$2. They sell all the candy bars at a price of two for $1.\$1. What was their profit, in dollars?

  2. Problem 2

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    A positive number xx has the property that x%x\% of xx is 4.4. What is x?x?

  3. Problem 3

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    Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

  4. Problem 4

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    At the beginning of the school year, Lisa’s goal was to earn an A on at least 80%80\% of her 5050 quizzes for the year. She earned an A on 2222 of the first 3030 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

  5. Problem 5

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    An 88-foot by 1010-foot floor is tiled with square tiles of size 11 foot by 11 foot. Each tile has a pattern consisting of four white quarter circles of radius 12\dfrac12 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

  6. Problem 6

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    In △ABC,\triangle ABC, we have AC=BC=7AC = BC = 7 and AB=2.AB = 2. Suppose that DD is a point on line ABAB such that BB lies between AA and DD and CD=8.CD = 8. What is BD?BD?

  7. Problem 7

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    What is the area enclosed by the graph of ∣3x∣+∣4y∣=12?|3x| + |4y| = 12?

  8. Problem 8

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    For how many values of aa is it true that the line y=x+ay = x + a passes through the vertex of the parabola y=x2+a2?y = x^2 + a^2?

  9. Problem 9

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    On a certain math exam, 10%10\% of the students got 7070 points, 25%25\% got 8080 points, 20%20\% got 8585 points, 15%15\% got 9090 points, and the rest got 9595 points. What is the difference between the mean and the median score on this exam?

  10. Problem 10

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    The first term of a sequence is 2005.2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 20052005th term of the sequence?

  11. Problem 11

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    An envelope contains eight bills: 22 ones, 22 fives, 22 tens, and 22 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20\$20 or more?

  12. Problem 12

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    The quadratic equation x2+mx+n=0x^2 + mx + n = 0 has roots that are twice those of x2+px+m=0,x^2 + px + m = 0, and none of m,m, n,n, and pp is zero. What is the value of np?\dfrac{n}{p}?

  13. Problem 13

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    Suppose that 4x1=5,4^{x_1} = 5, 5x2=6,5^{x_2} = 6, 6x3=7,6^{x_3} = 7, …,\ldots, 127x124=128.127^{x_{124}} = 128. What is x1x2⋯x124?x_1 x_2 \cdots x_{124}?

  14. Problem 14

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    A circle having center (0,k),(0, k), with k>6,k \gt 6, is tangent to the lines y=x,y = x, y=−xy = -x and y=6.y = 6. What is the radius of this circle?

  15. Problem 15

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    The sum of four two-digit numbers is 221.221. None of the eight digits is 00 and no two of them are the same. Which of the following is not included among the eight digits?

  16. Problem 16

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    Eight spheres of radius 1,1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

  17. Problem 17

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    How many distinct four-tuples (a,b,c,d)(a, b, c, d) of rational numbers are there with alog⁡102+blog⁡103+clog⁡105+dlog⁡107=2005? \begin{aligned} &a\log_{10} 2 + b\log_{10} 3 \\ &\quad {}+ c\log_{10} 5 + d\log_{10} 7 = 2005? \end{aligned}

  18. Problem 18

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    Let A(2,2)A(2, 2) and B(7,7)B(7, 7) be points in the plane. Define RR as the region in the first quadrant consisting of those points CC such that △ABC\triangle ABC is an acute triangle. What is the closest integer to the area of the region R?R?

  19. Problem 19

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    Let xx and yy be two-digit integers such that yy is obtained by reversing the digits of x.x. The integers xx and yy satisfy x2−y2=m2x^2 - y^2 = m^2 for some positive integer m.m. What is x+y+m?x + y + m?

  20. Problem 20

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    Let a,a, b,b, c,c, d,d, e,e, f,f, gg and hh be distinct elements in the set {−7,−5,−3,−2,2,4,6,13}. \{-7, -5, -3, -2, 2, 4, 6, 13\}. What is the minimum possible value of (a+b+c+d)2+(e+f+g+h)2? \begin{aligned} &(a + b + c + d)^2 \\ &\quad {}+ (e + f + g + h)^2? \end{aligned}

  21. Problem 21

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    A positive integer nn has 6060 divisors and 7n7n has 8080 divisors. What is the greatest integer kk such that 7k7^k divides n?n?

  22. Problem 22

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    A sequence of complex numbers z0,z_0, z1,z_1, z2,z_2, …\ldots is defined by the rule zn+1=iznzn‾, z_{n+1} = \dfrac{i z_n}{\overline{z_n}}, where zn‾\overline{z_n} is the complex conjugate of znz_n and i2=−1.i^2 = -1. Suppose that ∣z0∣=1|z_0| = 1 and z2005=1.z_{2005} = 1. How many possible values are there for z0?z_0?

  23. Problem 23

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    Let SS be the set of ordered triples (x,y,z)(x, y, z) of real numbers for which log⁡10(x+y)=z \log_{10}(x + y) = z and log⁡10(x2+y2)=z+1. \log_{10}(x^2 + y^2) = z + 1. There are real numbers aa and bb such that for all ordered triples (x,y,z)(x, y, z) in SS we have x3+y3=a⋅103z+b⋅102z.x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. What is the value of a+b?a + b?

  24. Problem 24

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    All three vertices of an equilateral triangle are on the parabola y=x2,y = x^2, and one of its sides has a slope of 2.2. The xx-coordinates of the three vertices have a sum of mn,\dfrac{m}{n}, where mm and nn are relatively prime positive integers. What is the value of m+n?m + n?

  25. Problem 25

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    Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?

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.