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2005 AMC 10A

All 25 problems from the 2005 AMC 10A. 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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    While eating out, Mike and Joe each tipped their server $2.\$2. Mike tipped 10%10\% of his bill and Joe tipped 20%20\% of his bill. What was the difference, in dollars, between their bills?

  2. Problem 2

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    For each pair of real numbers a≠b,a \neq b, define the operation ⋆\star as (a⋆b)=a+ba−b. (a \star b) = \frac{a+b}{a-b}. What is the value of ((1⋆2)⋆3)?((1 \star 2) \star 3)?

  3. Problem 3

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

  4. Problem 4

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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?

  5. Problem 5

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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?

  6. Problem 6

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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?

  7. Problem 7

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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?

  8. Problem 8

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    In the figure, the length of side ABAB of square ABCDABCD is 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?

  9. Problem 9

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    Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?

  10. Problem 10

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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?

  11. Problem 11

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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?

  12. Problem 12

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    The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length 2?2?

  13. Problem 13

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    How many positive integers nn satisfy the following condition: (130n)50>n100>2200? (130n)^{50} \gt n^{100} \gt 2^{200}?

  14. Problem 14

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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?

  15. Problem 15

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    How many positive cubes divide 3!⋅5!⋅7!?3! \cdot 5! \cdot 7!?

  16. Problem 16

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    The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6.6. How many two-digit numbers have this property?

  17. Problem 17

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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 this 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 this order. What is the middle term of the arithmetic sequence?

  18. Problem 18

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    Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?

  19. Problem 19

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    Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45∘,45^\circ, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point BB from the line on which the bases of the original squares were placed?

  20. Problem 20

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    An equiangular octagon has four sides of length 11 and four sides of length 22,\dfrac{\sqrt{2}}{2}, arranged so that no two consecutive sides have the same length. What is the area of the octagon?

  21. Problem 21

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    For how many positive integers nn does 1+2+⋯+n1 + 2 + \cdots + n evenly divide 6n?6n?

  22. Problem 22

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    Let SS be the set of the 20052005 smallest positive multiples of 4,4, and let TT be the set of the 20052005 smallest positive multiples of 6.6. How many elements are common to SS and T?T?

  23. Problem 23

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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?

  24. Problem 24

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    For each positive integer m>1,m \gt 1, let P(m)P(m) denote the greatest prime factor of m.m. For how many positive integers nn is it true that both P(n)=nP(n) = \sqrt{n} and P(n+48)=n+48?P(n + 48) = \sqrt{n + 48}?

  25. Problem 25

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    In △ABC\triangle ABC we have AB=25,AB = 25, BC=39,BC = 39, and AC=42.AC = 42. Points DD and EE are on ABAB and ACAC respectively, with AD=19AD = 19 and AE=14.AE = 14. What is the ratio of the area of triangle ADEADE to the area of the quadrilateral BCED?BCED?

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.