Competition · AMC preparation · step 4 of 4

AMC 12 2018B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2018B #1 grade 3+ arithmetic
Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. Ho…
AMC 12 2018B #2 grade 6+ rate-ratio
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his ave…
AMC 12 2018B #3 grade 8+ rate-ratio
A line with slope 2 intersects a line with slope 6 at the point (40,30). What is the distance between the x-intercepts o…
AMC 12 2018B #4 grade 8+ geometry-2d
A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of…
AMC 12 2018B #5 grade 7+ counting
How many subsets of {2,3,4,5,6,7,8,9} contain at least one prime number?
AMC 12 2018B #6 grade 7+ rate-ratio
Suppose S cans of soda can be purchased from a vending machine for Q quarters. Which of the following expressions descri…
AMC 12 2018B #7 grade 11+ algebra
What is the value of log₃7·log₅9·log₇11·log₉13…log₂₁25·log₂₃27?
AMC 12 2018B #8 grade 8+ geometry-2d
Line segment AB is a diameter of a circle with AB = 24. Point C, not equal to A or B, lies on the circle. As point C mov…
AMC 12 2018B #9 grade 6+ algebra
What is sum¹⁰⁰_(i=1) sum¹⁰⁰_(j=1) (i+j) ?
AMC 12 2018B #10 grade 6+ arithmetic
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct…
AMC 12 2018B #11 grade 8+ geometry-2d
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapp…
AMC 12 2018B #12 grade 10+ geometry-2d
Side AB of △ ABC has length 10. The bisector of angle A meets BC at D, and CD = 3. The set of all possible values of AC…
AMC 12 2018B #13 grade 10+ geometry-2d
Square ABCD has side length 30. Point P lies inside the square so that AP = 12 and BP = 26. The centroids of △ABP, △BCP,…
AMC 12 2018B #14 grade 6+ rate-ratio
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1…
AMC 12 2018B #15 grade 7+ number-theory
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
AMC 12 2018B #16 grade 12+ geometry-2d
The solutions to the equation (z+6)⁸=81 are connected in the complex plane to form a convex regular polygon, three of wh…
AMC 12 2018B #17 grade 7+ number-theory
Let p and q be positive integers such that 5/9 < p/q < 4/7and q is as small as possible. What is q-p?
AMC 12 2018B #18 grade 4+ algebra
A function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+nfor all integers n ≥ 3. What is f(2018)?
AMC 12 2018B #19 grade 6+ number-theory
Mary chose an even 4-digit number n. She wrote down all the divisors of n in increasing order from left to right: 1,2,…,…
AMC 12 2018B #20 grade 8+ geometry-2d
Let ABCDEF be a regular hexagon with side length 1. Denote by X, Y, and Z the midpoints of sides AB, CD, and EF, respect…
AMC 12 2018B #21 grade 10+ geometry-2d
In △ABC with side lengths AB = 13, AC = 12, and BC = 5, let O and I denote the circumcenter and incenter, respectively.…
AMC 12 2018B #22 grade 11+ algebra
Consider polynomials P(x) of degree at most 3, each of whose coefficients is an element of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9…
AMC 12 2018B #23 grade 11+ geometry-3d
Ajay is standing at point A near Pontianak, Indonesia, 0° latitude and 110° E longitude. Billy is standing at point B ne…
AMC 12 2018B #24 grade 8+ algebra
Let ⌊ x ⌋ denote the greatest integer less than or equal to x. How many real numbers x satisfy the equation x² + 10,000⌊…
AMC 12 2018B #25 grade 10+ geometry-2d
Circles ω₁, ω₂, and ω₃ each have radius 4 and are placed in the plane so that each circle is externally tangent to the o…

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