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2001 AMC 12

All 25 problems from the 2001 AMC 12. 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 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?

  2. Problem 2

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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?

  3. Problem 3

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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?

  4. Problem 4

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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?

  5. Problem 5

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    What is the product of all positive odd integers less than 10,000?10{,}000?

  6. Problem 6

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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.

  7. Problem 7

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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?

  8. Problem 8

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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?

  9. Problem 9

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    Let ff be a function satisfying f(xy)=f(x)yf(xy) = \dfrac{f(x)}{y} for all positive real numbers xx and y.y. If f(500)=3,f(500) = 3, what is the value of f(600)?f(600)?

  10. Problem 10

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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

  11. Problem 11

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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?

  12. Problem 12

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

  13. Problem 13

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    The parabola with equation y=ax2+bx+cy = ax^2 + bx + c and vertex (h,k)(h, k) is reflected about the line y=k.y = k. This results in the parabola with equation y=dx2+ex+f.y = dx^2 + ex + f. Which of the following equals a+b+c+d+e+f?a + b + c + d + e + f?

  14. Problem 14

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    Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9,A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set {A1,A2,…,A9}?\{A_1, A_2, \ldots, A_9\}?

  15. Problem 15

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    An insect lives on the surface of a regular tetrahedron with edges of length 1.1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)

  16. Problem 16

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    A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

  17. Problem 17

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    A point PP is selected at random from the interior of the pentagon with vertices A=(0,2),A = (0, 2), B=(4,0),B = (4, 0), C=(2π+1,0),C = (2\pi + 1, 0), D=(2π+1,4),D = (2\pi + 1, 4), and E=(0,4).E = (0, 4). What is the probability that ∠APB\angle APB is obtuse?

  18. Problem 18

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    A circle centered at AA with a radius of 11 and a circle centered at BB with a radius of 44 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is

  19. Problem 19

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    The polynomial P(x)=x3+ax2+bx+cP(x) = x^3 + ax^2 + bx + c has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the yy-intercept of the graph of y=P(x)y = P(x) is 2,2, what is b?b?

  20. Problem 20

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    Points A=(3,9),A = (3, 9), B=(1,1),B = (1, 1), C=(5,3),C = (5, 3), and D=(a,b)D = (a, b) lie in the first quadrant and are the vertices of quadrilateral ABCD.ABCD. The quadrilateral formed by joining the midpoints of AB‾,\overline{AB}, BC‾,\overline{BC}, CD‾,\overline{CD}, and DA‾\overline{DA} is a square. What is the sum of the coordinates of point D?D?

  21. Problem 21

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    Four positive integers a,a, b,b, c,c, and dd have a product of 8!8! and satisfy ab+a+b=524,ab + a + b = 524,bc+b+c=146,bc + b + c = 146,cd+c+d=104.cd + c + d = 104. What is a−d?a - d?

  22. Problem 22

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    In rectangle ABCD,ABCD, points FF and GG lie on AB‾\overline{AB} so that AF=FG=GBAF = FG = GB and EE is the midpoint of DC‾.\overline{DC}. Also, AC‾\overline{AC} intersects EF‾\overline{EF} at HH and EG‾\overline{EG} at J.J. The area of rectangle ABCDABCD is 70.70. Find the area of triangle EHJ.EHJ.

  23. Problem 23

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    A polynomial of degree four with leading coefficient 11 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

  24. Problem 24

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    In triangle ABC,ABC, ∠ABC=45∘.\angle ABC = 45^\circ. Point DD is on BC‾\overline{BC} so that 2⋅BD=CD2 \cdot BD = CD and ∠DAB=15∘.\angle DAB = 15^\circ. Find ∠ACB.\angle ACB.

  25. Problem 25

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    Consider sequences of positive real numbers of the form x,x, 2000,2000, y,y, …,\ldots, in which every term after the first is 11 less than the product of its two immediate neighbors. For how many different values of xx does the term 20012001 appear somewhere in the sequence?

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.