Competition · AMC preparation · step 4 of 4

AMC 12 2017A: 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 2017A #1 grade 4+ arithmetic
Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2 each, and 5-…
AMC 12 2017A #2 grade 7+ algebra
The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
AMC 12 2017A #3 grade 6+ number-theory
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A…
AMC 12 2017A #4 grade 8+ geometry-2d
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east…
AMC 12 2017A #5 grade 4+ counting
At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know…
AMC 12 2017A #6 grade 7+ geometry-2d
Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3 cm, 7…
AMC 12 2017A #7 grade 4+ algebra
Define a function on the positive integers recursively by f(1) = 2, f(n) = f(n-1) + 1 if n is even, and f(n) = f(n-2) +…
AMC 12 2017A #8 grade 8+ geometry-3d
The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume 216π. What i…
AMC 12 2017A #9 grade 8+ geometry-2d
Let S be the set of points (x,y) in the coordinate plane such that two of the three quantities 3, x+2, and y-4 are equal…
AMC 12 2017A #10 grade 7+ probability
Chloe chooses a real number uniformly at random from the interval [0, 2017]. Independently, Laurent chooses a real numbe…
AMC 12 2017A #11 grade 8+ geometry-2d
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 2017. She then discov…
AMC 12 2017A #12 grade 4+ number-theory
There are 10 horses, named Horse 1, Horse 2, . . . , Horse 10. They get their names from how many minutes it takes them…
AMC 12 2017A #13 grade 7+ rate-ratio
Driving at a constant speed, Sharon usually takes 180 minutes to drive from her house to her mother's house. One day Sha…
AMC 12 2017A #14 grade 4+ counting
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the fiv…
AMC 12 2017A #15 grade 11+ algebra
Let f(x) = sinx + 2cosx + 3tanx, using radian measure for the variable x. In what interval does the smallest positive va…
AMC 12 2017A #16 grade 10+ geometry-2d
In the figure below, semicircles with centers at A and B and with radii 2 and 1, respectively, are drawn in the interior…
AMC 12 2017A #17 grade 12+ algebra
There are 24 different complex numbers z such that z²⁴=1. For how many of these is z⁶ a real number?
AMC 12 2017A #18 grade 6+ number-theory
Let S(n) equal the sum of the digits of positive integer n. For example, S(1507) = 13. For a particular positive integer…
AMC 12 2017A #19 grade 8+ geometry-2d
A square with side length x is inscribed in a right triangle with sides of length 3, 4, and 5 so that one vertex of the…
AMC 12 2017A #20 grade 11+ algebra
How many ordered pairs (a,b) such that a is a positive real number and b is an integer between 2 and 200, inclusive, sat…
AMC 12 2017A #21 grade 9+ algebra
A set S is constructed as follows. To begin, S = {0,10}. Repeatedly, as long as possible, if x is an integer root of som…
AMC 12 2017A #22 grade 10+ probability
A square is drawn in the Cartesian coordinate plane with vertices at (2, 2), (-2, 2), (-2, -2), (2, -2). A particle star…
AMC 12 2017A #23 grade 8+ algebra
For certain real numbers a, b, and c, the polynomial g(x) = x³ + ax² + x + 10has three distinct roots, and each root of…
AMC 12 2017A #24 grade 11+ geometry-2d
Quadrilateral ABCD is inscribed in circle O and has side lengths AB=3, BC=2, CD=6, and DA=8. Let X and Y be points on BD…
AMC 12 2017A #25 grade 12+ probability
The vertices V of a centrally symmetric hexagon in the complex plane are given by V={ √2i,-√2i, 1/√8(1+i),1/√8(-1+i),1/√…

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