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2001 AMC 10

All 25 problems from the 2001 AMC 10. 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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    The median of the list n, n+3, n+4, n+5, n+6, n+8, n+10, n+12, n+15 \begin{gathered} n,\ n+3,\ n+4,\ n+5,\ n+6, \\ \ n+8,\ n+10,\ n+12,\ n+15 \end{gathered} is 10.10. What is the mean?

  2. Problem 2

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    A number xx is 22 more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

  3. Problem 3

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    The sum of two numbers is S.S. Suppose 33 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

  4. Problem 4

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    What is the maximum number of possible points of intersection of a circle and a triangle?

  5. Problem 5

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    How many of the twelve pentominoes pictured below have at least one line of symmetry?

  6. Problem 6

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    Let P(n)P(n) and S(n)S(n) denote the product and the sum, respectively, of the digits of the integer n.n. For example, P(23)=6P(23)=6 and S(23)=5.S(23)=5. Suppose NN is a two-digit number such that N=P(N)+S(N).N=P(N)+S(N). What is the units digit of N?N?

  7. Problem 7

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    When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

  8. Problem 8

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    Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

  9. Problem 9

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    The state income tax where Kristin lives is levied at the rate of p%p\% of the first $28000\$28000 of annual income plus (p+2)%(p+2)\% of any amount above $28000.\$28000. Kristin noticed that the state income tax she paid amounted to (p+0.25)%(p+0.25)\% of her annual income. What was her annual income?

  10. Problem 10

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    If x,x, y,y, and zz are positive with xy=24,xy=24, xz=48,xz=48, and yz=72,yz=72, then x+y+zx+y+z is

  11. Problem 11

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    Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 88 unit squares. The second ring contains 1616 unit squares. If we continue this process, the number of unit squares in the 100100th ring is

  12. Problem 12

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    Suppose that nn is the product of three consecutive integers and that nn is divisible by 7.7. Which of the following is not necessarily a divisor of n?n?

  13. Problem 13

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    A telephone number has the form ABC−DEF−GHIJ,ABC-DEF-GHIJ, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, A>B>C,A\gt B\gt C, D>E>F,D\gt E\gt F, and G>H>I>J.G\gt H\gt I\gt J. Furthermore, D,D, E,E, and FF are consecutive even digits; G,G, H,H, I,I, and JJ are consecutive odd digits; and A+B+C=9.A+B+C=9. Find A.A.

  14. Problem 14

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    A charity sells 140140 benefit tickets for a total of $2001.\$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?

  15. Problem 15

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    A street has parallel curbs 4040 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 1515 feet and each stripe is 5050 feet long. Find the distance, in feet, between the stripes.

  16. Problem 16

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    The mean of three numbers is 1010 more than the least of the numbers and 1515 less than the greatest. The median of the three numbers is 5.5. What is their sum?

  17. Problem 17

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    Which of the cones below can be formed from a 252∘252^\circ sector of a circle of radius 1010 by aligning the two straight sides?

  18. Problem 18

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    The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

  19. Problem 19

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    Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

  20. Problem 20

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    A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length 2000.2000. What is the length of each side of the octagon?

  21. Problem 21

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    A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 1010 and altitude 12,12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

  22. Problem 22

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    In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by v,v, w,w, x,x, y,y, and z.z. Find y+z.y+z.

  23. Problem 23

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    A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?

  24. Problem 24

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    In trapezoid ABCD,ABCD, AB‾\overline{AB} and CD‾\overline{CD} are perpendicular to AD‾,\overline{AD}, with AB+CD=BC,AB+CD=BC, AB<CD,AB\lt CD, and AD=7.AD=7. What is AB⋅CD?AB\cdot CD?

  25. Problem 25

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    How many positive integers not exceeding 20012001 are multiples of 33 or 44 but not 5?5?

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.