Competition · AMC preparation · step 4 of 4

AMC 12 2017B: 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 2017B #1 grade 8+ rate-ratio
Kymbrea's comic book collection currently has 30 comic books in it, and she is adding to her collection at the rate of 2…
AMC 12 2017B #2 grade 7+ algebra
Real numbers x, y, and z satisfy the inequalities 0<x<1, -1<y<0, and 1<z<2. Which of the following numbers is necessaril…
AMC 12 2017B #3 grade 7+ algebra
Supposed that x and y are nonzero real numbers such that (3x+y)/(x-3y)=-2. What is the value of (x+3y)/(3x-y)?
AMC 12 2017B #4 grade 7+ rate-ratio
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had…
AMC 12 2017B #5 grade 6+ arithmetic
The data set [6, 19, 33, 33, 39, 41, 41, 43, 51, 57] has median Q₂ = 40, first quartile Q₁ = 33, and third quartile Q₃ =…
AMC 12 2017B #6 grade 8+ geometry-2d
The circle having (0,0) and (8,6) as the endpoints of a diameter intersects the x-axis at a second point. What is the x-…
AMC 12 2017B #7 grade 11+ algebra
The functions sin(x) and cos(x) are periodic with least period 2π. What is the least period of the function cos(sin(x))?
AMC 12 2017B #8 grade 9+ algebra
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagon…
AMC 12 2017B #9 grade 10+ geometry-2d
A circle has center (-10, -4) and has radius 13. Another circle has center (3, 9) and radius √65. The line passing throu…
AMC 12 2017B #10 grade 7+ rate-ratio
At Typico High School, 60% of the students like dancing, and the rest dislike it. Of those who like dancing, 80% say tha…
AMC 12 2017B #11 grade 7+ counting
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either…
AMC 12 2017B #12 grade 12+ algebra
What is the sum of the roots of z¹²=64 that have a positive real part?
AMC 12 2017B #13 grade 8+ geometry-2d
In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two…
AMC 12 2017B #14 grade 8+ geometry-3d
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inc…
AMC 12 2017B #15 grade 8+ geometry-2d
Let ABC be an equilateral triangle. Extend side AB beyond B to a point B' so that BB'=3 · AB. Similarly, extend side BC…
AMC 12 2017B #16 grade 7+ probability
The number 21!=51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random. Wh…
AMC 12 2017B #17 grade 7+ probability
A coin is biased in such a way that on each toss the probability of heads is 2/3 and the probability of tails is 1/3. Th…
AMC 12 2017B #18 grade 8+ geometry-2d
The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD=3. Point E is chosen so t…
AMC 12 2017B #19 grade 6+ number-theory
Let N=123456789101112…4344 be the 79-digit number that is formed by writing the integers from 1 to 44 in order, one afte…
AMC 12 2017B #20 grade 11+ probability
Real numbers x and y are chosen independently and uniformly at random from the interval (0,1). What is the probability t…
AMC 12 2017B #21 grade 6+ algebra
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. Aft…
AMC 12 2017B #22 grade 11+ probability
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four roun…
AMC 12 2017B #23 grade 11+ algebra
The graph of y=f(x), where f(x) is a polynomial of degree 3, contains points A(2,4), B(3,9), and C(4,16). Lines AB, AC,…
AMC 12 2017B #24 grade 11+ geometry-2d
Quadrilateral ABCD has right angles at B and C, △ ABC ~ △ BCD, and AB > BC. There is a point E in the interior of ABCD s…
AMC 12 2017B #25 grade 11+ counting
A set of n people participate in an online video basketball tournament. Each person may be a member of any number of 5-p…

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