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2005 AMC 10B

All 25 problems from the 2005 AMC 10B. 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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    A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

  4. Problem 4

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    For real numbers aa and b,b, define a⋄b=a2+b2.a \diamond b = \sqrt{a^2 + b^2}. What is the value of (5⋄12)⋄((−12)⋄(−5))?(5 \diamond 12) \diamond ((-12) \diamond (-5))?

  5. Problem 5

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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?

  6. Problem 6

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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?

  7. Problem 7

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    A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

  8. Problem 8

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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?

  9. Problem 9

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    One fair die has faces 1,1, 1,1, 2,2, 2,2, 3,3, 33 and another has faces 4,4, 4,4, 5,5, 5,5, 6,6, 6.6. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

  10. Problem 10

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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?

  11. Problem 11

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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?

  12. Problem 12

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    Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

  13. Problem 13

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    How many numbers between 11 and 20052005 are integer multiples of 33 or 44 but not 12?12?

  14. Problem 14

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    Equilateral △ABC\triangle ABC has side length 2,2, MM is the midpoint of AC‾,\overline{AC}, and CC is the midpoint of BD‾.\overline{BD}. What is the area of △CDM?\triangle CDM?

  15. Problem 15

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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?

  16. Problem 16

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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}?

  17. Problem 17

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    Suppose that 4a=5,4^a = 5, 5b=6,5^b = 6, 6c=7,6^c = 7, and 7d=8.7^d = 8. What is a⋅b⋅c⋅d?a \cdot b \cdot c \cdot d?

  18. Problem 18

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    All of David’s telephone numbers have the form 555–abc–defg,555\text{–}abc\text{–}defg, where a,a, b,b, c,c, d,d, e,e, f,f, and gg are distinct digits and in increasing order, and none is either 00 or 1.1. How many different telephone numbers can David have?

  19. Problem 19

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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?

  20. Problem 20

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    What is the average (mean) of all 55-digit numbers that can be formed by using each of the digits 1,1, 3,3, 5,5, 7,7, and 88 exactly once?

  21. Problem 21

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    Forty slips are placed into a hat, each bearing a number 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, 7,7, 8,8, 9,9, or 10,10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let pp be the probability that all four slips bear the same number. Let qq be the probability that two of the slips bear a number aa and the other two bear a number b≠a.b \ne a. What is the value of qp?\dfrac{q}{p}?

  22. Problem 22

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    For how many positive integers nn less than or equal to 2424 is n!n! evenly divisible by 1+2+⋯+n?1 + 2 + \cdots + n?

  23. Problem 23

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    In trapezoid ABCDABCD we have AB‾\overline{AB} parallel to DC‾,\overline{DC}, EE as the midpoint of BC‾,\overline{BC}, and FF as the midpoint of DA‾.\overline{DA}. The area of ABEFABEF is twice the area of FECD.FECD. What is ABDC?\dfrac{AB}{DC}?

  24. Problem 24

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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?

  25. Problem 25

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    A subset BB of the set of integers from 11 to 100,100, inclusive, has the property that no two elements of BB sum to 125.125. What is the maximum possible number of elements in B?B?

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 10 archive.