Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 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 10 2018B #3 grade 3+ algebra
In the expression ( × )+( × ) each blank is to be filled in with one of the digits 1,2,3, or 4, with each digit being us…
AMC 10 2018B #4 grade 8+ geometry-2d
A three-dimensional rectangular box with dimensions X, Y, and Z has faces whose surface areas are 24, 24, 48, 48, 72, an…
AMC 10 2018B #5 grade 7+ counting
How many subsets of {2,3,4,5,6,7,8,9} contain at least one prime number?
AMC 10 2018B #6 grade 7+ probability
A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the…
AMC 10 2018B #7 grade 7+ geometry-2d
In the figure below, N congruent semicircles lie on the diameter of a large semicircle, with their diameters covering th…
AMC 10 2018B #8 grade 4+ arithmetic
Sara makes a staircase out of toothpicks as shown: This is a 3-step staircase and uses 18 toothpicks. How many steps wou…
AMC 10 2018B #9 grade 7+ probability
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let p be the probabilities that when all…
AMC 10 2018B #10 grade 8+ geometry-3d
In the rectangular parallelepiped shown, AB = 3, BC = 1, and CG = 2. Point M is the midpoint of FG. What is the volume o…
AMC 10 2018B #11 grade 6+ number-theory
Which of the following expressions is never a prime number when p is a prime number?
AMC 10 2018B #12 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 10 2018B #13 grade 6+ number-theory
How many of the first 2018 numbers in the sequence 101, 1001, 10001, 100001, … are divisible by 101?
AMC 10 2018B #14 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 10 2018B #15 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 10 2018B #16 grade 6+ number-theory
Let a₁,a₂,…,a₂₀₁₈ be a strictly increasing sequence of positive integers such that a₁+a₂+…+a₂₀₁₈=2018²⁰¹⁸. What is the r…
AMC 10 2018B #17 grade 8+ geometry-2d
In rectangle PQRS, PQ=8 and QR=6. Points A and B lie on PQ, points C and D lie on QR, points E and F lie on RS, and poin…
AMC 10 2018B #18 grade 7+ arithmetic
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy th…
AMC 10 2018B #19 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 10 2018B #20 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 10 2018B #21 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 10 2018B #22 grade 8+ geometry-2d
Real numbers x and y are chosen independently and uniformly at random from the interval [0,1]. Which of the following nu…
AMC 10 2018B #23 grade 7+ number-theory
How many ordered pairs (a, b) of positive integers satisfy the equation a· b + 63 = 20· lcm(a, b) + 12·gcd(a,b), where g…
AMC 10 2018B #24 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 10 2018B #25 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⌊…

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